You can do a 3 way deal if everyone cooperates. If you want to buy a bike from dealership A and dealership B wants your car, you get dealership B to write a cheque for the price of the car to dealership A. You go back to dealership A with dealership B's cheque and the remaining balance and dealership A can process it as a trade and save you the tax.
I have done it before, but it's probably easier (and more cost effective) to sell the car privately and cleanly buy the bike.