Start with an American Standard Cadet 3. It is a very good toilet, but also small and (crucially) cheap.
I would then screw it on to a wooden platform that you had attached axles and wheels to, perhaps like a go-cart and preferably with steering.
Next, find a racey toilet seat. Attach.
Drill a hole in the wooden platform under the toilet, and perhaps fit a flange, so that when you fill the tank and flush, the waste will come out of here. Fashion an exhaust and attach to the hole/flange and shape so the waste pops out from the rear of the cart. 4 inch PVC pipe would look "racey".
Use a garden hose and an adaptor to screw onto the fill valve inlet to "refuel" with water when empty.
A lot of members on this forum use these techniques for their home/clients' plumbing, although often with less prestigious toilets, so they will no doubt have other ideas.