Inquisitors are funner. They have more skills and Bane is extremely strong. They are very similar though.
I think there would be break points where one would do better than the other. And there are open questions like, does the chaplain have access to advanced weapon training. Also gear will matter, does the chaplain have gloves of dueling?
They'll both be fine.
What do you prefer: Feats or Skills?
The latter is tops for DPR, all else being equal. They're both pretty good though.
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The Arsenal Champion.
Fervor (to cast self-buffs), Sacred Weapon, and Weapon Training slightly outpace Studied Target and Bane. Particularly since Weapon Training and the damage die increase of Sacred Weapon don’t take any actions. Sneak attack can bring that number closer but it’s hard to get sneak attacks at range.
But the real cherry on top is the combat feats that the Warpriest gets. Especially since she counts as full BAB and full fighter levels for those bonus feats.
Is this a character you're going to play or are you just getting ideas?
If it's for play then it probably doesn't matter which you pick, they'll both make strong DPR characters - likely stronger than is necessary.
If it's for a competition with a friend or something then I'm sure peoople can post some builds.
I'm torn because Arsenal Chaplian is a fighter with less BAB but 6th level spells (which more than makes up for the loss of BAB). That many feats is so fun to play with. Also prepared Divine casters get every spell which is so useful sometimes.
On the other hand I hate 2+INT skills (unless I'm playing an INT-based caster). I also generally prefer spontaneous casting if all else is equal just for the flexibility in the moment. I have to spend more time planning spells while building the character, but less time searching through spell-tables while actually playing.
I'd say skills vs feats and spontaneous vs prepared will be more impactful than DPR.
Arsenal Chaplain would likely do more damage. Inquisitor is probably the second runner up in terms of archer damage, but has better skills.
The Slayer is by no means bad at archery, but the Inquisitor's Bane ability and Judgements really allow for great damage bonuses, as does the Arsenal Chaplain's ability to self buff as a quick action with Divine Power/Favor. The biggest edge the Arsenal Chaplain has over the Inquisitor is extra feats which makes the Arsenal Chaplain build come online earlier, and they get access to Point Blank Master which the Inquisitor does not.
Personally, I've played both the Arsenal Chaplain and the Inquisitor (no archetype) and a Ranger archer (which has some similarities to the Slayer) and my personal favorite was the Inquisitor.
For the Inquisitor I played a feather domain Inquisitor, which allowed me to get an animal companion and be a turret of death. You don't need Point Blank Master when you're a small sized race mounted on your medium companion which ferries you everywhere. Don't forget that you can make full attacks while mounted even if you mount moves. You don't take any penalties to your attacks even unless they move twice, but you still get to make all your attacks. The Inquisitor also had many more skills than the Warpriest. And while the ability to access any Cleric spell of level 6 or lower is nice, I found the spontaneous nature of the Inquisitor's spell casting better suited my tastes as I was able to select the most important spells of each level that I would want to use often, and keep wands/scrolls of ones that were useful but not needed everyday. I also did look at myself to be the cure/bandaid for the party so I didn't consider status removal to be important to my role, but I could attempt it with scrolls/wands.
Honestly, the Inquisitor just offers so much flexibility that to me it is the most fun to play. It doesn't perform as well at archery as the Warpriest, but it's almost as good, has much better skills, and I prefer spontaneous spell casting because I can pick a handful of spells I generally want to have available over preparing exactly the number of Divine Favor spells I will need for the day.
Edit: Just realized you specifically asked about the Inquisitor archetype Sanctified Slayer not the Slayer class. I want to specify I didn't play the Sanctified Slayer archetype and would like to say I consider it bad for the build. Why? Because Sneak Attack doesn't work well with ranged builds. Normally you have to be within 30ft, which is the antithesis of what I want to do with a ranged build. The second part is it gives up judgement for studied target. Studied Target is good, and starts off better than Judgement. Eventually however Judgement provides the same bonuses (maybe +-1) but has options beyond just attack and damage and eventually can use up to 3 of those options like the attack bonus, damage bonus, and the fast healing bonus. So ultimately I think Judgement is stronger in the long run. So if you plan to play a game at high levels (8th level is when Inquisitor can have two Judgements at same time) or if the game is a one shot at low levels it can help inform you decision about what will be better.
Studied Target has two big advantages. First, it has infinite uses per day, so you can use it for every encounter. Second, it applies to some prime inquisitor skills in or out of combat.
The action economy on Studied Target is weird. It can be done before combat in some cases, but if you are caught off guard you will need to use a combat action to use it, at level 7 it's only a swift action which is on par with judgement.
I'd say Studied Target is easily the easier ability to run. You know what your buffs are going to be or not be for every fight. And you don't lose anything by studying everything and everyone you come across. It's hard to say which one is better, but Studied Target certainly has its high points.
Note: I agree that the sneak attack is a bit of a waste on a ranged build. You'll mostly only get it on the first round of combat. And the Sanctified Slayer gets its slayer talents oddly late, negating one of the talent's main draws, early combat feats through combat styles.
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Looking solely at "who is the better ranged combatant," the Arsenal Champion's edge is action economy. Weapon Training doesn't take an action and Fervorcasting is swift. There are a few breakpoints where the inquisitor has the edge, but the last one is really at 7th level (earlier if the MAC gets Gloves of Dueling earlier) Let's compare what happens with two 7th level characters and two 9th level characters:
MAC - weapon training always applies, swift action fervor divine favor, full attack
+3 to hit, +3 to damage
SS - Swift action study, full attack
+2 to hit, +2 to damage
Round 2:
MAC - swift action Sacred Weapon, full attack
+4 to hit, +4 to damage
SS - Swift Action bane, full attack
+4 to hit, +4+2d6 to damage
If the SS had a chance to study the target ahead of time, then she can skip straight to round 2. She has the advantage here.
Round 1:
MAC - weapon training always applies, swift action fervor divine favor, full attack
+7 to hit, +7 to damage
SS - Swift action study, full attack
+2 to hit, +2 to damage
Round 2:
MAC - swift action Sacred Weapon, full attack
+9 to hit, +9 to damage
SS - Swift Action bane, full attack
+4 to hit, +4+2d6 to damage
The Inquisitor does have an option to narrow the gap a bit, and that's to give up a round to cast a self-buff spell. So in this 9th level example
MAC - weapon training always applies, swift action fervor divine favor, full attack
+7 to hit, +7 to damage
SS - Swift action study, standard action cast divine favor, move action to do whatever
No damage done this round
Round 2:
MAC - swift action Sacred Weapon, full attack
+9 to hit, +9 to damage
SS - Swift Action bane, full attack
+7 to hit, +7+2d6 to damage
So it's quite a bit closer and if the target is easy to hit the inquisitor will be doing more damage from here on out. But she had to give up an entire round of damage to get there.
10: SS picks up +1 atk/damage from studied target
11: nobody gets a damage boost
12: MAC picks up +1 atk/damage from Sacred Weapon, SS picks up +2d6 damage from greater bane
13: MAC picks up +1 atk/damage from Weapon Training
14: no change
15: SS picks up +1 atk/damage from studied target
16: MAC picks up +1 atk/damage from Sacred Weapon
17: MAC picks up +1 atk/damage from Weapon Training
From 8th level on the MAC is getting +2 atk/damage every 4 levels (between Weapon Training and Sacred Weapon). The SS is getting +1 atk/damage every 5 levels. And the MAC is probably fervorcasting divine favor or a similar buff every combat without losing a full-attack. Then you consider the combat feats, which I didn't even include in those examples.
That's not to say the SS doesn't have some advantages. The SS is a better buffer prior to combat (heroism isn't even on the warpriest spell list), has a better initiative, and gets those nice teamwork feats. But overall the MAC has the edge.
warpriest bonus feats has 'Finally, for the purposes of these feats, the warpriest can select feats that have a minimum number of fighter levels as a prerequisite, treating his warpriest level as his fighter level."
the Arsenal archtype gain weapon training ".. At 5th level, a Molthuni arsenal chaplain gains weapon training as per the fighter class feature"
which mean that beside the general weapon focus\specialization\greater etc, he can pick up the following feat :
advanced weapon training
with which he can take the adv, weapon training of focused weapon (that give him back the lost warpriest sacred weapon level 6+ weapon damage).
and more impotently he can also nab the adv. weapon training of warrior spirit which can give him a very long duration of any bane ability (and more) that he would need to get on per with the inquisitor (and i think maybe even longer duration overall then the inquisitor - min vs rounds)
edit : with sacred weapon and spirit warrior ability to increase the weapon total enchantment bonus. getting a +1, beaming weapon and making it count for a the few moments of activating the orbs as +5 will net you 5 X searing lights (caster level 10) to use as a free action during that day.
the warprist can pull it off with a +2 worth weapon, the inquisitor would need a +6 weapon (+5, beaming) to pull this off.
on a side note, if you do take weapon focus as an option with the advanced weapon training feat (and especially if you take it at the first level it is open -6) i would fully understand if your GM nuke your character sheet from orbit.
'oh i take an archetype that give up any increase to my sacred weapon damage beyond 1d6 but on level 5 he get weapon training'
...level 6 - 'i use by bonus feat to regain my increase to my sacred weapon damage.'
- so you actuly just gave up a bonus feat for that weapon training, didn't you?
Advanced Weapon Training is not valid for Arsenal Chaplain's in PFS so it's debatable whether any gm would allow it or if it's even possible especially considering PFS's track record with balance.
As for combat power, Sneak Attack is actually halfway decent on the Sanctified Slayer who wins Initiative frequently and also has the abilities and skill ranks to identify and ambush threats before they are aware.
In pure numbers, the Arsenal Chaplain wins. I believe in actual gameplay, the Inquisitor is the better ranged combatant.
Advanced Weapon Training is not valid for Arsenal Chaplain's in PFS so it's debatable whether any gm would allow it or if it's even possible especially considering PFS's track record with balance.
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if you actuly look at what it say it say :
"The Molthuni Arsenal Chaplain does not qualify for the Advanced Weapon Training on pages 18-19."well if you actuly check the book the words "advanced weapon training" are used for 2 things. the option of a fighter to take different abilities instead of gaining more weapon groups in weapon training - at pages 18-19
and the feat im talking about - the feats start at page 20+. i know aon mark it as page 19, but the first feat in this book Ace Disarm is in page 20.
i think who ever in aon linked the pages to the feat maybe made a hasty search and got the fighter option page instead.
anyway. while i agree that for pfs the fighter swipe option is illegal WITHOUT TAKING THE FEAT, as the warpriest doesn't get any advanced weapon groups to swipe with the ability. he can qualify for the feat getting the option that way.
i would agree though that warrior spirit is pfs illegal anyway. but in the first hand i never wrote (nor did the op asked) about pfs legal rules in specific.