Well I know that Ubuntu provides you with the option to install proprietary drivers, I do not know about Linux Mint. But maybe you just have to enable third party programs in your package manager? I'd do a search for ATI in your package manager, and be sure to search ALL available packages, not only supported ones or open source ones. Maybe someone who has installed Linux Mint can give you advanced tips on that.
As far as I know, the performance of the proprietary drivers is better, but the openGL support of proprietary ATI drivers is troublesome. Nevertheless it's worth a try. Good luck!