Those are called canister Thrones.You could get a keller shell and wrap it. I see them on ebay all the time.Then you could cut out 2 pieces of 3/4" plywood the would fit snuggly into the shell. Fit one into the bottom of the shell and attach the other to the seat and set that on top of the shell. You could even cover the bare plywood with felt for aesthetic reasons, but you have to allow for the thickness of the felt on the fit, or just paint it black. If you covered the bottom in felt it would peel back sooner or later.
Yeah... canister thrones.
I'd probably also stick some adjustable feet on the bottom to level it on uneven stages. I run into those from time to time. Like the feet on the bottom of appliances like washers/dryers and refrigerators.