Can you put a bayonet on a Buckler Gun?
Yes, you could, but then you wouldn't be able to fire it. Also, since the bayonet is a two-handed weapon, it would be awfully awkward to wield properly (maybe represented by carrying over the "always considered an off-hand weapon" limitation of the buckler itself).
Neither. Bayonet counts as both a Light Blade and as a Close weapon.
Trying to free up that other hand?
Not planning to reload during a fight then I take it? I would just use a pirate hook hand along with the buckler gun. That way you can make melee attacks, ranged attacks, and avoid any penalties whatsoever.
Hook Hand, from Pirates of the Inner Sea
10gp, 1 lb., Simple Light Melee Weapon, 1d4/x2 slashing damage, Disarm quality
Alternatively, use some other cup-type of weapon such as Ultimate Combat's scizore or pata if you are looking for more damage.
If you are wearing heavy armor, just bolt a locked gauntlet onto it in place of a traditional gauntlet and then lock your cup-weapon of choice to it.
Are you missing a hand, or the whole arm? If you are missing more than the hand, I'm not sure how you intend to use a buckler.
Why not just put them on your good arm? As far as I can tell, it doesn't even use up your hand.
It seems not (it being a two-handed weapon). Guess you will have to settle for blade boots or armor spikes.