Realistically any chassis will do for autocross because it's not the car that makes you fast (at least for the 1st 2-4 years) it's you who makes the car fast.

Longer wheel bases like coupes and sedans are more stable under hard braking and cornering. Whereas hatchbacks are snappy - meaning get ready to work out your sphincter

Start out with a EG Civic Si or the 2nd gen CRX Si. Both good platforms to start out with. The 99-00 Civic Si might get bumped into HS which I think would do pretty well in that class against the Mini Cooper.

But remember mod the driver or nut behind the wheel before the car.