My experience has been that curse doesn't require a particularly specific party combination and they should be fine. (Currently running book six).
It's heavily urban focused but for a single book, so there's limited use for wilderness focused abilites. Druids and Rangers won't find much to do. There's no particular point where Arcanists are needed, although always useful, and the city provides convenient access to scrolls for the most part. A use magic device character could work for any arcana specific problems.
Book 5 -requires- a cleric, or somebody who can handle heavy stat drain, fight a lot of undead, cast certain cleric spells, etc. However recognising this the module does provide a helpful NPC cleric at the time, so this is pretty taken care of if the party doesn't screw up and get them killed (or kill them or refuse to work with them)
A trapbreaker is helpful, particularly for Book 3.
Broadly speaking, fights happen in enclosed small environments - little to mid sized urban rooms - which limits area effects, flight, and ranged combat in effectiveness.(all still usable, but we've had a few fights with people stuck in doorways).
Moreover you have 5 PCs to the modules expected 4, so encounters will generally be easier across the board unless you tune them up.
Aside from having no arcane casting besides bard (which is just a buff specialist), you have a heavy dependence in Wisdom as an ability score and boosters that would be helped by having someone move to Int or even another Cha score dependency. While Reverse is correct regarding some specific use Arcane magic being handled by scroll, ensuring you have the right scrolls for any circumstance can become expensive, especially when you are forced to leave the city. I also disagree with his comment about druid's and rangers: not only is there an urban archetype for both classes, but Ranger has an archetype specifically dedicated to Korvosa (Sable Company Marine, or Sky Stalker on d20pfsrd).
Also, by book 5, the Warpriest (probably not the Inquisitor because he shouldn't waste a spell known on this) should be able to cast all the spells necessary for Scarwall, even without the help of the provided Cleric.
Another concern I have is that there is a pretty good character that can join the party as a cohort starting in book 4, but she is also a bard, so it might be too redundant to have them and an actual bard if they are just looking for good buffs but can wait a while to get them.
Lastly, while this otherwise could work as a good party line up, you have to consider that Slayer, Warpriest, and Inquisitor can all be melee or ranged builds perfectly interchangeably. Bard and Shaman really could to but they would likely opt to backline. Be sure advise them against cluttering the front line too much and stepping on each other's toes, or the opposite and having too thin a front line with everyone in the back being constantly at risk.
My best suggestion would probably be having the bard switch to a more martial focus class (Fighter, Paladin, Ranger, Barbarian, etc.) so at least one of the Slayer/Warpriest/Inquisitor can drop back to a backline/ranged role while the others offer buffing or other support in a more midfielding position to shore up the Full BAB class. It also means if someone really likes that bard in the game already, you don't have to modify her to make her useful to the team.
There are a lot of hard to hit high hp monsters in this campaign as well, I have two dedicated melee combatants (slayer and swashbuckler)and one melee based cleric. The cleric spent a lot of feats and has a high strength on trying to be a good fighter but only good base attack bonus makes a huge deal. He just does not dish out the damage that a fighter type class can deal.
That said my group is a slayer, swashbuckler/shadowdancer, sorcerer and cleric. A really good group that deals a lot of damage and solves a lot of problems well. But their cleric being so melee focused cannot keep up with the healing demands of the party. That is my worry with that group.
The one player is an Investigator not a inquisitor
Blasted dyslexia.
In any case, if the slayer and Warpriest are frontline that's pretty good. I will point out that Warpriest actually doesn't have that much healing, their "channel equivalent" ability Fervor is fairly limited (especially since it doesn't scale by ability score and doesn't have an "Extra Fervor" feat) but also consumes spell slots as normal, so unless he is wasting the entire point of Warpriest (the self buffing on top of limited cleric utility), he will likely not have much left over for healing. At least Paladin Lay-on-Hands has to be used on healing (or damaging undead, which is a terrible alternative given they can save), but the choice is ultimately up to them.
For the investigator, unless he takes infusion, he isn't really an arcane problem solver, and even then (with or without) he sorta fills the same role as the Warpriest: combat (self-)buffing. So arcane hurdles can still become full-stop obstacles. I will also add that if he doesn't want to be melee, there is a feat called Ranged Study that lets him drop to midfield (30ft range increment) and can be taken before he even gets studied combat at level 4.
Only thing left is Shaman and Bard, but they can pick up the slack of Warpriest along the healing front because as mentioned above, in practice Warpriest tends to do less healing.