You will find it difficult to hire a classic Mini but should be able to find a MINI to hire. Beaulieu show always has lots of new MINI's on display as the Mini Cooper Register who organise Beaulieu welcome ALL Mini's so no problem there. The MCR member who runs the Beaulieu Cooper Day is also a keen MINI and Mini owner and enthusiast.
Cheers for the heads up, looking forward to seeing all the Minis / MINIs / mini skirts / Mini-Me impersonators / mini golf...