Since the advent of JAR over a decade ago the old 5,700Kg limit has disappeared. A Spitfire is an SEP(L) class aeroplane and you can fly it with a PPL, CPL or ATPL which includes a valid SEP(L) rating so long as you have the relevant differences signed off. That means Tailwheel, VP Prop, Retractable and Supercharged engines (Also pressurisation if you are flying some later marks).
There is no requirement for a "type rating exemption" unless flying an aeroplane which would normally require a type rating; that means a jet or a multi crew aircraft type.
A multi engined warbird such as the P38 or a mosquito (I wish!) would require an MEP rating.
That's the legal answer but of course considerable training would also be required to satisfy both the owner and his insurance company!
Happy landings
3 Point