Surely K had to know that J would use the weapon he was given. Is there any reason he would give him the Noisy Cricket, a weapon that can blow huge holes in the side of semi truck trailers? Was he trying to teach him that things aren't always what they seem? (i.e. size =/= strength) It seems like they would want to try and keep incidents like that to a minimum.
My impression was that although Kay evidently had quite a high impression of Jay overall, he felt that his attitude could use some serious adjustment. There are various jibes about his age and his mental immaturity culminating in Kay refusing to let him have (or even use) the neuralyser.
When they're about to face off with the Bug, he can't seem to bring himself to give Jay one of the big guns but at the same time he's professional (and self-interested) enough to recognise that refusing him a weapon entirely would be grossly stupid, since he would expect Jay to be covering his back against an especially nasty alien.
In the end he settles for the the next best thing, giving Jay a gun that looks puny (for the comedy value) but actually packs a pretty solid punch. He gets to emphasise who's the senior partner in the outfit but still has the security of having Jay carry a fairly decent weapon.
In the official film novelisation, after using it, even Jay acknowledges that the name is pretty damn funny.
" 'Noisy Cricket,' huh? Very funny. Ho, ho, ho." Jay looked at the tiny gun.