I would say that if you are going to do it just put a little bit more money into the build and buy an old rear engine rider for it's transmission. They can sell for very little or free sometimes, take the trans from that and put it in the mower. This is what I am doing and have almost 600 dollars into my mower and still need a few more parts. Whatever you do it is going to cost you way more than what you originally budgeted. I have budgeted 100 dollars of mine when I first started it. these projects get way outta hand fast. If you do not believe me just ask Feerlessfront he has way more into his racer/ go kart project.
My advice is to run a peerless 700 three or five speed with a chain drive to a solid axle setup, from what I have heard the machine will be much more fun to drive. I cannot verify this because I have a little bit more work to do before I drive mine.