It does'nt have to be only cheapo tyres that can cause a problem, I had Goodyear Eagle F1 tyres on my 2wd Saff and they were really bad at tramlining. It felt like someone else suddenly took control of the car and it steered itself. I changed them for a set of road legal track day tyres and it transformed the handling with no more tramlining. Maybe the Michelin tyres just dont suit the car? I would check underneath all the suspension components and if they are all ok the tyres would be my next suspect. The way you have described how it handles just seems so odd for such a low mileage car.