First, some self-imposed rules:
Some items may not be equipped. This includes the Infinity Helm and, if obtained, the *Light* *Sword*.
Classes that get Stealth are disallowed.
For this playthrough, stamina casters are disallowed.
(Note that this means there's nobody to cover Locks and Traps, but having multiple characters with Knock Knock should allow me to deal with locked doors when necessary.)

Already decided for this party:
Priest (battle priest with Mace/Shield, should I raise Piety or ignore it?)
Mage (Powercast, of course)
Alchemist (Staff of Doom + Powercast)
Psionic (Slings; maybe my Communications character since I will have no Bard?)
Bishop

Anyway, two questions:
In this party, which spellbooks should my Bishop focus on? (Keep in mind that every spell that one character needs to know is covered by one of the other casters.)
Who should be the 6th character? Fighter? Or should I take a hybrid?

The question is: why would you take bishop when there are already four focused casters? OR rather, why take four focused casters when you can take one bishop?

Alternative party idea (within your self-imposed rules):
4 Mook Fighters, each with Giant Sword
1 Elf Bishop
1 Hobbit Samurai

It is a nice theme. Any other character would break it. Unless it is another caster, e.g. another Bishop.
However: you do not need to pick 6 characters so why force it?
Obviously it would be a huge gain in power to pick a Fighter. It's only that it is a lot more interesting w/o that.
If you want that however I'd say it should most definitely be a Fighter, not a Hybrid.

On a side note I don't know why you want the Staff of Doom on the Alchemist. I'd say the Mage would be more fitting. For one thing the Alchemist can cast Death Cloud anyway and for another the Mage is more dependent on protection and can't use Shields, which is a good reason for a long ranged two handed Staff.

I'd take a lizard gadit who fights with a sword and not cast spells but thats prob too far from the box... also my bishop (dog) would go priest spells because the l7 book hits very hard with the bishop powercast skill that a priest doesn't norm get.

I am actually seriously considering Fighter simply because I don't use that class that much, and with this party I wouldn't need to worry about not having enough portals. One issue with the Fighter, however, is the inability to use the Lance, which is one of the better early game weapons.

If you are recommending the Mage for the Staff of Doom, what race would you suggest that would get Power Cast early and Power Strike at a reasonable level? What other spellbook should I pair with priest for the Bishop? (I would go priest anyway because it gives me another character who can cast Restoration eventually.)

I should probably mention that it may be a while before I actually try this party, as I may want to play some other games first (Undertale, try to finish Wizardry Gaiden 3, maybe some action game, etc.).

I tend to go physic spells with my Bishop because that makes for a strong defence | support character. Mage | Alchemist spells look good on paper but the average Bishop takes a long time to build and always feels underpowered as a damage caster… mind stab, pandemonium and a late game sane mind. The truth is any spell book has potential so its largely “what spells you like best”.

Priest:
Yeah, I think you can ignore PIE at least for a while. STR+INT looks like a good build to me

Bishop:
I would not focus on any of the spellbooks, but rather build up all of them and use his spells more as support (similar to hybrids). Instead I would try to make him a bit more effective in combat and possible utilize his Artifacts skill (i.e. scrolls, magical weapons etc.). The thing is, no matter which books you focus on, he will always be worse than four your specialists.

6th character:
I would build some tank (i.e max out VIT for iron skin). I myself tried my walkthrough with only the 4 casters, and at times this was what I lacked somewhat. A Fighter seems obvious for this, but if you like Lances or Polearms in general, a Valkyrie might be reasonable too (though you don't really need another magic user).


Btw: One thing I am trying out in some minor modding attempt is a different approach for a Fighter (I even renamed him Barbarian), which is: Even higher VIT (and maybe also STR) requirement, higher base HP (10 sounds fine), he gets damage resistance (instead of may KO opponents), but in turn cannot wear heavy armor (similar to a Rogue, but without Stealth of course). Instead of Close Combat he gets an Axe bonus (Close Combat I gave to the Lord instead).
I am sort of curious how high HP and damage resistance (from class and later iron skin) compares to high AC (base AC or from armor).

If I ignore PIE on this character, the build would probably be STR+DEX. My Bishop should be able to handle any powercasting of divine spells I need.

I actually did STR+INT on a Priest once; I ended up changing him into a Lord later on.

I could note that, in my experience, Bishops get more spell points than other classes thanks to all the spells they can learn from books and the availability of multiple spellbook skills.

Also, this party, when I run it, will be all female like my other recent parties.

(As a side note, I am glad that Wizardry 8 Bishops are actually good for once; in older Wizardries, Bishops were rather weak until very high levels (25+).)

That's kinda tricky indeed. I tend to decide on which stats to minimize so the others will be good in due time. For a Mage I'd say Senses and Vitality ought to be minimized. His HP pool is pretty low and the VIT % boni won't get him very far. He is a dedicated melee with the Staff of Doom so Senses is redundant pretty much. Speed is needed (although not maxed any time soon) and Piety is also nice for SP, Dexterity is required.
You could go for an Elf due to his low VIT and SEN. Then again you only "save" 15 points so you might as well go for a Human, which will have Powerstrike a little sooner. That's what I'd do I guess. With all Spells available Resistances should be fine anyway. That said there is no ideal Powerstrike Mage race I'd say: none is overly specialized for the task. I like Humans as casters actually. My last Mage was a Human :)

Power Cast @11 & Powerstrike @16 if you go for those from the getgo. That's a tad late for the Power Cast, which you'd like to have when you grind (if you grind) but still within limits. Level 16 should be hit before you go "late game" at least, either the Rapax or Bayjin I mean, so that's okay I'd say. Dexterity would take 11 levels to max so you could either delay Strength to 26 for a level 12 INT/DEX powerspike or go for level 22 max DEX after INT and STR. Of course you could forego DEX in favor of SPE if you max INT/STR first, in case you fancy initiative.