Well, Toon Link is slower than Bowser in air, and he's pretty light. The main issue is, he's a weapon/projectiles style enemy. The sword gives him extra reach to boost his priority. And unlike Link, he's not sluggish.
Use the Firebreath to chew up early damage, intermixed with ducking swipes. Try jumping at him and blowing fire. Eventually he'll start blocking it as you come in, and that's your cue to jump-and-klaw.
You generally don't want him above you. Rather than mess with his Bombs and Sword pogo, just roll away and make some room. Instead, get above him and hash him up.
Try to attack from the corners. This is not so easy for him to strike; he'll have to rely on his boomerang, which takes too long, or come up after you/move. Forcing the enemy to move is generally a good thing. When he starts hopping up after you and doing Up+B, jump back and blow flame.
If he gets aggressive, fight back with your own agression. Hop-maneuver and blow flame in his face, occasionally popping him with your claws. If you catch him open under you, bomb him.
He will try to stay at sword's end in distance from you... show him that is not a safe distance using fire and jump attacks, and he will come in closer. Then you can nail him.