[Wereboar]

For whatever reason, my bishop, 3 hybrids, gadgeteer and bard just can't get the job done.

You tried a very slow developing party. Hybrids will become quite strong, but are quite weak until level ~12 in my experience.
Bishop will get many good spells, but levels slowly. In addition, since he doesn't get bonus to the realms, he'll have problems to reach skill 30 until level 5 (and 45 for 8), to be able to learn the better spells. (this is for 'expert' setting; in normal, skills raise much faster through use, so it probably is no problem)
The gadgeteer gets very powerful very late, but is weak until his omnigun gets repeater, and you have better ammo.
Bard is the only fast developing character in your party; at least if you use him for music mainly, not for fighting.

The one thing I see this party missing is someone I can rely on to throw bombs, etc., when the time comes. I'm trying to build up the priest and mage through combat practice, but it is slow going.

You don't need to throw bombs. I rarely do it; only when using a rogue with cursed weapons. Most other classes have some ranged ability.

Don't waste time to train your spellcasters as fighters. The mage will use spells nearly all of the time. Use a sling early; later, you'll rarely need it.
Learn 'make wounds' with the priest. I didn't in early games, to save the spell points for heal spells. But the attack spell raises the skills very fast.

Unrelatedly, I am considering taking this party to the Umpani camp before Arnika (to get some modern weapons and ammo, and some of the better spellbooks for my mage).

Arnika has many good spell books, too. And you shouldn't go for modern weapons with a fighter IMO. The advantage of modern weapons is that you don't need high strength for good damage, but you fighters will build up strength anyway. And don't listen to talk about the phaser; it's easy to get when reloading, but rare otherwise (< 10% in my experience), so you probably won't find it with your playing style.