Its the transmission, why would an engine mount "start working" after the car is warm?
I have heard the same thing about not changing the fluid. The stuff in the fluid is helping the transmission function properly... Allegedly. I haven't got too much experience with that personally, but i've
heard the same so there might be some truth in it.
secondly, you work at a sports car course and you still went for the auto? (even after all the stuff?) Come on man!
5spd swap ftw.