Tamiya M-03 Mini tyres & TT01 question

[General Accident]

Any suggestions for a decent tyre for a Mini on carpet. The kit ones suck and the bloody thing goes round corners like its got castors on instead of wheels. Spinning is an understatement. I tried to fit some wheels off a Schumacher Mi3 onto the Mini but they fouled the hubs so I'm guessing I'll have to stick with the Tamiya wheels.

Also, will "proper" premounted tyres ie, RP's or similar fit onto a TT01 chassis car, I've seen they go for cheap money so I fancied a dabble but if I can't get half-decent rubber I'm not going to bother.