Cannon
While in theory this might seem no better than "Gun", I find that it's generally used in a more versatile manner to indicate things that fire projectiles other than bullets. A Fire Cannon would naturally be a device which shoots fire. An Axel Cannon might lead some to think it's a typo for something that shoots axles, but is still clearly a weapon such that things come out of it to strike a target at range. It's used for non-physical projectiles as well, with Laser Cannons being popular in fiction and Sound Cannons (or Sonic Cannons) being real and legitimate weapons that do pretty much what one would expect from the name. The only downside I can think of is that it's so popular that some combinations are already associated strongly with a more specific concept, so a cannon that shoots hands might not be clear if you call it a Hand Cannon unless you're intentionally making a pun. I would consider this the best and most general option, that seems most likely to replace shooter everywhere you need it. Another option though...
Launcher
While I wouldn't call it universal, "Launcher" is a word used alongside a projectile to indicate "something that shoots that projectile". Notable examples include Rocket Launcher, a device which shoots rockets, and Grenade Launcher, a device which shoots grenades. I don't know that it truly fits your needs since I would find it somewhat awkward to simply replace "shooter" in your examples, but it is nicely unambiguous: a fire launcher is very unlikely to be anything other than a device that fire comes out of.
Battery
You identified this, and it could be useful on occasion. In particular, a Battery should be shooting a lot of things at once. Your spell that shoots a lot of fireballs could be referred to as a "Flame Battery". It would be somewhat awkward tough, batteries are much more likely to be a fixed placement rather than something you can pull out as needed, and it doesn't feel right in the name for a spell. As I said, it could be usable on rare occasions.
Verb
s', or 'artifact/machine which is used toVerb
with'. Potholders are a tool, and so are screwdrivers and typewriters. But lienholders are people, and so are busdrivers and ghostwriters. It all depends on which sense got hold first.