What sort of effects do you want the bike to have? If all you need the physics for is gravity, jumps and collisions then you could manually tilt the bike according to how fast you are turning - I can help you with the physics equations there if you like - while using things like gravity to determine the position of the entire bike in space as a single entity. You could even do things like wheelies this way, but it will involve a lot of computation by yourself. The added benefit is that this would probably only require one physics entity, so you will only need the professional version.

What sort of context will you be using it in? This would be fairly easy to implement on a superbike race course, harder to implement on a dirt-bike track and difficult to do for a city street.

Rephrased (I hope) more clearly, you would use the physics engine to keep the bike on the ground and prevent collisions, but you would use your own code to tilt and turn the bike. Of course, if you wanted a really spectacular crash, you would need to enable physics for each part of the bike, rather than as a whole, but you could do that the instant the rider looses control of the bike.