I finally switched out the chain and sprockets between the motor and transaxle for a timing belt and some pulleys. Although it's much smoother, I'm starting to think I still won't get as much speed as I could. The transaxle came from a mobility scooter and obviously wasn't designed to go 20mph, and it's starting to show. I haven't bought the 36V battery pack to test the whole drive system on full-power so I still have some hope, but I think there's a good chance I'll have to find a replacement that can do what I want.

A new rear axle would also most likely give me some more space for a brake system that doesn't have to sit so close to the wheels. Currently, there's almost no room for the tires to flex.