I didn't mean to say that an 18" is totally impossible but it might be a bit of a challenge getting the carb on however the engine itself would probably fit. You might have to use an offset intake manifold like the ones Dax has in their parts section on page 2. It will make the carb stick out a little bit on the side to make it work. Spoke count is also a factor. You'll want 36 spoke rims as opposed to 32 if you plan to use the kit rag joint or if you get a sprocket adapter of most aftermarket adapters. I would say, trade in the bike for one with a bigger frame.
I wouldn't recommend a road bike unless you can get different rims with wide tires to fit on it. Those thin tires will burn out quick and provide much with less traction but people have motorized road bikes.