This reply is a bit late now but in case anyone else has a handling problem with a mk3 MX-5:
DO NOT PUT LOW BUDGET TYRES ON THE REAR !!!!
You will get away with it on the front (up to a point) but not on the rear.
The performance will be almost unbelievably poor! It is really shocking at how much difference it makes to the handling.
I certainly agree that the recommended Michelins are used.