As I understand it, there can be problems with holding carbureted bikes in long wheelies. It has to do with the float bowl being held at the wrong angle with no additional forces acting to keep the fuel in the bowl properly.

Double check the FZ1 to see if it's actually fuel injected. If it is, I can't imagine any other reason why it wouldn't be as good a stunter as any other sportbike. The FZ1's were tuned for a broader power curve and more torque, with translates into smoother power delivery which I imagine would be helpful for stunting.

All that said, a dual sport bike or motard seems like a better choice yet. Being based on a dirtbike it should stand up to abuse better than I'd expect a sportbike to.

Good luck finding a new toy!