i will say, the classes seem very strong. they all get a lot of skills (the inquisitor and slayer get FIVE, which is crazy because that's more then a rogue), and the inquisitor gets extra attack AND sneak attack-esque radiant damage AND up to 7th level spells. i think other then skills the other classes are fine, but i wonder if inquisitor might be a tad overtuned. i guess castigation is supposed to be balanced by not being able to castigate certain creature types, but that feels more like the class is neutered if you're fighting those enemies. i wonder if it'd be more balanced if you could either castigate OR use extra attack? that way castigate is your main since it does more damage and fuels your judgement pool, but extra attack is there as a fall back if you're fighting something you can't castigate. this is just a first impression, though, so maybe it all works better in actual play.
also it's weird that castigation doesn't work with ranged weapons yet the inquisitor iconic has a pistol.
So... the Inquisitor's Castigation:
Yeah, it's like sneak attack damage. And at level 1, it's the same level as a sneak attack, but you can apply it more liberally at that level range and don't need to rely on enemy positioning or advantage. But it also starts falling behind at 3rd level, and you need a bonus action to perform your Absolvement, meaning your option for off-hand attacks as a fallback for rounds where you miss your one attack is less of an option. (Not not an option, just less of one)
By level 5 you're dropping 1d8+3d6+Dex as a rogue and 1d8+2d6+Dex as an Inquisitor. To offset the throughput difference, you get an extra attack which also offsets the unavailability of bonus action striking as a fallback for missing and skipping out on damage for the round. Which puts the throughput to 1d8+4d6+Dex for Rogue and 2d8+2d6+(Dex(2)) for an Inquisitor. But your healing pull is still capped at (1d8+2d6+Dex)/2. So it kinda works out.
However, the gap widens continually across all following levels. Yeah, the Inquisitor eventually gets 2d8+7d6+(Dex(2)) as a throughput option on their turn, but the Rogue's swinging 1d8+11d6+Dex. Assuming they don't go for d8s through the Burglar archetype, for which the Inquisitor has no equivalent.
And then comes the limited creature typing to further bring it below average overall output compared to a Rogue while still fulfilling much of the same class identity (big hits, skills, etc) -and- being the party healer. In those fights it behooves you to aim not at the dogs but at the huntsman for your hits to try and keep your pool high.
There's also the question of what's -on- the spell list. There -are- 5 damaging spells, but the majority of the spell list is either "Set up someone else with your action which slows down your damage and healing capabilities" or utility. So they'll mostly, but not entirely, be used out of combat.
As far as it not working with ranged attacks: Yeah. That's intentional. I wanted there to be situations where you can't easily use it in a given situation. As for why Stalvos has a pistol: It's more about the dramatic threat of training a pistol on someone as you recite their sins and heresies. Holding someone at the point of your rapier is plenty dramatic, but there's just something about putting a gun to someone's head that has a harsher vibe to it for visual imagery on your intimidation checks!
As far as the Slayer and Inquisitor both having a pile of skills: That's also intentional. I've always hated the "There's one person in the party with a bunch of skills and everyone else gets two or three" dynamic. I'm much more enamored of White Wolf's "Everyone gets a bunch of skills, but not all of them" structure so there's some overlap between concepts and for the most commonly used skills (Perception, Stealth, Etc)