Optimizing any spellcaster
The most powerful thing in the game you can possibly get is always “the next level of spells.” Ensuring you get that next spell level as soon as possible is step 1 to optimizing any spellcaster.¹ Simply being an arcanist is already questionable, just because (starting at 3rd level) your spellcasting is a level behind a wizard (an arcanist gets 2nd-level spells at 4th; a wizard gets them at 3rd). Missing more spellcasting levels for any reason is always going to be questionable.
For a topical example, D&D 3.5e—which Pathfinder is based on—offered a prestige class called the malconvoker (Complete Scoundrel). The malconvoker missed spellcasting at 1st level, but otherwise had a full spellcasting progression. (D&D 3.5e also didn’t have anything like Prestigious Spellcaster to make up the missing level.) It was focused on summoning evil creatures, and their summon monster spells summoned twice as many evil monsters for twice as long, effectively making them 4× as good (sort of; you rarely really get a lot of use out of the doubled duration on summons). That made the malconvoker maybe worth considering the missed spellcasting—but it was a debatable choice. That is how strong a spellcasting level is.
On the other hand, it must be said from the outset, there is room to be suboptimal here. The wizard is a fantastically overpowered class, and the arcanist is only a bit weaker²—which means it is still pretty overpowered. Plus Prestigious Spellcaster exists—a feat is also a lot to pay, but it’s absolutely worth getting that spellcasting.
On summoning in general
The most important thing for any summoner to do is somehow, someway reduce the 1-round casting time of summoning spells to a full-round action (or less). A 1-round casting time means that you do nothing only turn, and you are extremely vulnerable for an entire round (if you get hit, you almost certainly will lose the spell), and you only get the summon for your next turn. That is really rough, and really not what you want to be doing.
Some tables just houserule it to be a full-round action—or simply misunderstand the rules and think a 1-round casting time is a full-round action. That’s the best thing, since it costs you nothing. Failing that, you’ll want Sacred Summons—which your build gets from blackfire adept, but that is immensely painful prior to 6th level. You also have Quicken Spell, which works, but is far too pricey for this purpose (better to buy a quicken metamagic rod, IMO).
Arcanist has the weaker spell progression of a sorcerer, but gets to switch up their spells daily like a cleric or wizard, making them somewhere in between the two classes. Ultimately, cleric or wizard is stronger because they get each spell level from 2nd on a class level earlier, and that’s huge. Also, the exploiter wizard exists, so if you really want certain exploits, you can get them.
On the other hand, the occultist is pretty good. The planar ally spells are pretty strong, contact other plane is very strong and you get it way early, and conjurer’s focus allowing you to cast summon monster on-time goes a certain way towards mitigating the delay on your other spells—plus, bonus, they’re standard actions when cast this way, which is a big deal. However, the limitation of only one summon monster spell at a time is significant here, and ultimately your arcane reservoir is a lot more limited than your spell slots, so you won’t be summoning as much this way. But it’s enough to get you to 6th level and Sacred Summons.
You might want to consider the wizard class, specifically the Familiar Folio pact wizard archetype (not the Haunted Heroes archetype of the same name). In addition to the spellcasting that’s a level ahead, the pact wizard gets an aura, Sacred Summons as a bonus feat at 1st level, and (eventually) an improved familiar for free. So it’s fairly similar to a blackfire adept, but without having to actually take the prestige class. You’re limited to a single subtype of summon, though, which is painful—choose carefully.
The other option—and probably the winner here—is the cleric, which (obviously) gets an aura, and can thus take Sacred Summons at any time. Moreover, there’s an archetype that’s really good for summoning: not the herald caller, which claims to be all about summoning but is kinda meh, but the evangelist, which gets bardic performance. Bardic performance is best when you have things to buff with it—and summoning ensures you will. Action conflict is tricky, but in any situation where you can pre-buff, this is golden.
This is a weird one. If you don’t have an aura, this is how you get Sacred Summons—that probably is worth the price of entry, but compared to a pact wizard or cleric, who could have just had Sacred Summons for 5 levels already, without having to burn two feats on Favored Prestige Class and Prestigious Spellcaster, it’s not looking like a winner. Blackfire pact is pretty good, for improving planar ally and/or binding, but this benefit is already a feat (Augment Calling), so you’re effectively paying 2 feats for what is otherwise 1 feat, so again, blackfire adept still looks poor compared to a pact wizard or cleric. (Augment Calling and blackfire pact do stack though, so that’s pretty nice.)
What might save blackfire adept—maybe—are the breach and breaching legion features. Those require another lost spellcasting level—read: another feat burned on Prestigious Spellcaster—but breach is a rather unique effect. It’s extremely situational—those effects are rather rare in my experience—but when you need it, nothing else will do. So it becomes a question of how often you’re going to use it. If it’s only once or twice a campaign, then giving up three feats is dubious. If it’s going to happen more than that, though, it could easily be worth it.
But unless I knew the campaign was going to feature a lot of dead magic zones or antimagic fields, I wouldn’t bother—I’d prefer cleric or wizard. If you’re set on the arcanist, though, then yeah, this is how you’d do it. Definitely take at least 8 levels (and a second iteration of Prestigious Spellcaster), though. I would strongly consider taking all 10 (and a third Prestigious Spellcaster), because riftward breaching is pretty good too.
Your build has a couple of critical mistakes with respect to this class:
You don’t actually qualify. You need Great Fortitude, and you don’t have it. I can’t blame you, I wouldn’t want to take Great Fortitude either, but it’s non-optional if you want to be a souldrinker.
You can’t—officially—take Favored Prestige Class twice. All feats are limited to only being taken once unless it has a Special section that says otherwise, and Favored Prestige Class doesn’t. That means you can’t use Prestigious Spellcaster to restore souldrinker’s lost spellcasting—and that makes the class absolutely awful for you.
The official rule here is stupid, and you could ask your GM for a houserule to let you take it more than once—which they should allow—but the official rule is the rule unless your GM agrees to do otherwise.
However, the point is kind of moot, even assuming you fix 1 and get a favorable houserule for 2: I’m not loving this class—particularly for you. Nothing here helps summoning, and you need to burn two more feats on Favored Prestige Class and Prestigious Spellcaster, plus you need Great Fortitude, which sucks. Charon evangelist is definitely The Way to Go™ here, so you’re right on how you want to use it, but to leverage that you would need to focus on item creation, not summoning.
I would much, much rather finish blackfire adept and have a feat left over than try to force this class into the build. And again, you need to take more levels of this to make your investment in it worthwhile—particularly with Charon, you want that second boon, but unless you take 6 levels of souldrinker—and therefore take Prestigious Spellcaster again—you’re not getting it any faster than anyone else who took Fiendish Obedience.
First of all, a note: your build doesn’t include Great Fortitude, which is required for souldrinker. So that’s an issue. Great Fortitude is a terrible feat, too, so having to take it is a pretty big drawback of that class. Worse, officially, you can’t take Favored Prestige Class in more than one prestige class, so you can’t get Prestigious Spellcaster for both blackfire adept and souldrinker.
Spell Focus (Conjuration)—probably useless to you (summons don’t have save DCs; actually, most conjurations of any stripe don’t), but required by a bunch of things you do want, so you’ll have to pay your taxes.
Augment Summoning—yes, it’s a good feat.
Quicken Spell—useless as early as you’re getting it. Eventually, quickening things is valuable, but it’s a very high-level trick unless you have some way to reduce or mitigate the spell level adjustment (which, as far as I can tell, you do not).
I would strongly consider whether or not I could just get away with a quicken metamagic rod here. They’re expensive, but then so is 4 spell levels.
Finally also note that quickening—whether with the feat or the rod—becomes much more important as an evangelist, since you can cast a quickened summon monster spell in the same turn you start inspire courage.
Superior Summoning—not great. Summoning a bunch of lower-level monsters is almost never what you want to do, and so getting one more when you do that isn’t exciting. You’ll rarely use it and the benefit isn’t that great even when you do.
Augment Calling—this is pretty great. Definitely get this. You might even want to take it more than once to hit more subtypes, depending. Stacks with the blackfire adept’s blackfire pacts, which is nice.
Favored Prestige Class—OK, ish, but not so much when you’re only taking 4 levels.
Prestigious Spellcaster—absolutely mandatory if you lose spellcasting to a prestige class. But a feat is also a significant cost. The blackfire adept is probably worth it. The souldrinker? Less convinced.
Fiendish Obedience—required for souldrinker, and not bad in its own right. Charon evangelist is pretty much the Correct™ use of the feat for one, too, so you’re right about that. But you need to do some item creation if you want to leverage Charon here.
So I would drop Superior Summoning, and either drop Fiendish Obedience, or take some item creation feats—probably Craft Wondrous Item, maybe Craft Magic Arms and Armor, maybe Craft Rod or Craft Staff. But that feels like it’s getting away from summoning, to me, though you can buy it back by crafting items that improve summoning.
I’d also probably drop souldrinker, even if allowed to make it work, and thus save three or more feats (Great Fortitude, Favored Prestige Class, and Prestigious Spellcaster). If I’m an arcanist, I’d give one—if not two—of those back by taking more Prestigious Spellcasters for blackfire adept, though I’m personally more likely to do this as a single-classed evangelist cleric (at which point I’m taking feats relating to bardic performance).
And, by extension, optimizing any character, since spellcasters are optimal by a huge margin.
Specifically, assuming they are the same level, the arcanist is a bit stronger for 60% of levels (when both arcanist and wizard have the same level of spells), but much weaker the other 40% of the time (when wizards have spells a level higher). See here for a full discussion.
Referring to the updated—nerfed—version of the class. There is an older version that is much, much more powerful, but since it’s out of date you usually won’t be allowed to play it. Also, shout-out to the create item ability from soul pool—that one is pretty good, especially for Charon.