You can remove the entire body and just bolt the 'spine' chassis with front and rear suspension subframes, engine and box into your own space-frame chassis and add a body of your own design/choosing.
That's effectively how the MEV Exocet is built (sadly most of the pics have been lost to Photobucket ransom demands, but enough remain to give you the idea) -
https://www.pistonheads.com/gassing/topic.asp?t=1152576
The resulting car will need to be IVA tested, so design and build it with the IVA regs in hand and budget the extra £500 or so to get it through.
The only plus side with that is that you at least don't suffer the ignominy of a 'Q' plate, but get a registration number related to the age of the donor MX5. You'll also have a unique car to your own design.