Eldritch Knight, maybe.
Evangelist does that for 9/10 of its levels. Eldritch Knight doesn't advance a familiar's abilities but it fits most of your other requirements.
A Witch with the Winter Witch Archetype can take the Winter Witch Prestige Class which grants full progression for all Witch Class Levels, (inc. Familiar progression), except 1 Level of Arcane Spellcasting. You can regain the most of the 1 lost Spell Level with Favoured Prestige Class and Prestigious Spellcaster if you have the Feat spaces, (this also helps out the Evangelist PrC) - you get the Caster Level progression but Prepared Arcane Spellcasters do not learn Spells from Level up from these Feats.
One awkward workaround for Familiar progression, (opening your needs up to just advancing Arcane Spellcasting) :
* At Character Level 3 take the Feat Familiar Bond.
* At Character Level 5 take the Feat Improved Familiar Bond -> your Familiar can have full progression (based on your H.D.).
N.B.: Selecting Human grants you a Bonus Feat that can hurry this up to Character Level 3.
N.B.: If you are in a home game that allows Templates, any that grant additional H.D. would actually advance your Familiar faster than your Class Level.
N.B.: If you only take 1-4 Levels in a PrC you can take the Feat Boon Companion to regain that lost Familiar progression - one Feat instead of three.
Searching quickly, I've found the Diabolist allows your familiar to continue advancing with you. The only downside to me though, is this class is evil.
Wouldn't Eldritch Knight progress the Familiar of an Eldritch Guardian Fighter? The Eldritch Knight levels count as Fighter levels, do they not?
Stargazer is definitely associated with a deity though an obscure one - 'Deity: Must worship Pulura.' There's background in the writeup on AoN about how some anti-demon stuff in the prestige class derives from the Worldwound swallowing up Sarkoris where Pulura's worship was centered, which may mean more to Golarion experts.
Maybe I am too lenient as a GM...
If you wanted to take two levels of the one Fighter archetype that gets a familiar, and 7-8 levels of Magus for 3rd level Arcane spells (with Magus Fighter Training and Imp. SpellCombat) for prerequisites into Eldritch Knight...
I would let it all stack/count, because it literally doesn't hurt anything to allow it to work how they want it to in order to get the character they want to play.
People get so caught up in the wording of things that they seem to forget this is a fantasy game.
Literally, what is so broken or unbalanced if you let the levels count as Fighter levels to continue the progression of your stupid Familiar?
It takes 5-7 levels in an Arcane spellcasting class to get 3rd level spells, anyways, so it's going to be 7-9 levels before you enter Eldritch Knight anyways.
Oh no, Spell Critical at level 19-20, with a Familiar that shares your Combat Feats, and you have SpellStrike and Imp. SpellCombat... that sounds exactly like what an Eldritch Knight is supposed to be.
One awkward workaround for Familiar progression, (opening your needs up to just advancing Arcane Spellcasting) :
There is a much less awkward method to get your familiar to advance. Eldritch Heritage[arcane] gives you a familiar with progression equal to your level-2. If you're a human you can take the trait Focused study which gives you skill focus at 1st level and several other levels. While normally you would be 2 levels behind taking two levels in fighter with eldritch guardian gives you 2 levels which stack with your level-2, effectively making it just equal to your level.
not only is this less feat intensive but you actually net some extra skill focus feats which you could spend on spellcraft or whatever else is a useful skill for your character.
Which familiar ability do you need? The familiar's hp, saves, skill bonuses, and maybe attack (not sure of that one off the top of my head) all advance as yours do, familiar-granting class or not.
A Witch with the Winter Witch Archetype can take the Winter Witch Prestige Class which grants full progression for all Witch Class Levels, (inc. Familiar progression), except 1 Level of Arcane Spellcasting. You can regain the most of the 1 lost Spell Level with Favoured Prestige Class and Prestigious Spellcaster if you have the Feat spaces, (this also helps out the Evangelist PrC) - you get the Caster Level progression but Prepared Arcane Spellcasters do not learn Spells from Level up from these Feats.
One awkward workaround for Familiar progression, (opening your needs up to just advancing Arcane Spellcasting) :
** spoiler omitted **...
Since the Evangelist prestige class does not have an explicit Spells per Day mechanic, the Prestigious Spellcaster feat is useless for getting back the caster level you lose at 1st level in that prestige class.
If you need only four levels out of the PrC, then Boon Companion will work with any of them.
Improved familiar options include resolute and entropic creatures as well as fiendish/celestial ones. Counterpoised seems reasonable...but I haven't seen it mentioned anywhere.