Fast Study is an amazing discovery. It permits you to leave spell slots open each day and then spend only 1 minute to write in any spell you need. Need an extra Fly spell? No problem. Need a Secure Shelter? No problem. Need an extra fireball for the upcoming fight against creatures weak to fire? No problem. It very nearly gives you the power to be spontaneous and is incredibly useful for the once in awhile spells you have lying around.
Disclaimer: I'm basing this line of thought on Pathfinder Society and the scenarios/modules ran in it. I don't care what you do in your homebrew setting or if you have an evil GM or any house rules you may have. I'm going strictly off of PFS rules.
There are several factors that come into play, none of which works in favor of your player.
Gunslingers are already amazingly powerful and overpowered (I know I'm going to get a lot of flak for that).
1. They can do touch AC attacks which will dominate almost anything at high levels that isn't specifically prepared for firearms (and few mythical creatures are). This means they can take all the penalties of deadly aim because they will virtually never miss, gaining a huge bonus to damage.
2. x4 critical and with improved critical that's a 19-20 range. A gunslinger can do 100+ damage with a single crit by level 5.
3. DEX to damage (I'm assuming he's a pistolero or a musket master). This is insane as it permits gunslingers to only need one stat for their ranged attacks. One stat that gives huge boosts to AC, initiative (to shoot flat-footed targets), reflex saves, damage, and to hit.
A musket master can only do a touch attack within 50 feet (if she spends a move-equivalent action to gain an extra 10 feet to the musket's range).
A rifle can shoot a touch attack up to 400 feet with penalties. However these penalties are very little because you're still shooting touch which is easy to hit and he probably will get far shot to lower the penalties.
If your player gets distance on his firearm, double the range. 800 feet for a Rifle and 100 feet for a musket. That's significant; that's sniper territory. I haven't even added the Deadeye Grit which only increases the range he can shoot at.
Imagine him shooting touch attacks at 1600 feet for a -15 penalty (not really that big at higher levels and its still just touch AC). This doesn't even include if he's got + weapon bonuses on his guns or buffs.
Example: Group sees a dragon in the sky far up and heading for the group, that gunslinger can probably kill it before it gets close.
The other main thing is that he will only be able to use metal cartridges for his gun, which will cost him 7.5 GP a shot (to make) unless he gets a potion/oil, wand, or a friendly caster just to cast abundant ammunition.
If you still want to do it (and I'm assuming he's a musket master), remind him that he needs to take Rapid Reload AGAIN as the rapid reload feat under musket master specifically says MUSKETS, not rifles.
There's virtually no advantage of going universalist: there will probably only be a handful of spells on the schools you prohibit and if you really want them, spend two spell slots. Also you can take opposition research at level 9, permitting you to remove this requirement.
I've always enjoyed flavor items and hope you guys put some more into the game.
I've felt that bards don't really have many interesting items that don't already duplicate another specific spell.
For example, the lyre of building is a very fun item that isn't specifically duplicated by another spell and is a great item for bard RP. Not to mention other items that are for the benefit of NPCs you may come across, such as a small table that enlarges into a buffet table to provide food for the group. I would like to see more items that support users can buy for the group's benefit.
Or a bard's music box that could hold a bard's song that he puts into it for a few rounds.