I have a friend who uses a lot of minkee for backs and she borders the minkee with 4-6" of cotton so her edges are stable and there is no bulk when she binds them with regular cotton.
I agree -- minkee really sheds -- the machine needs a major cleanup when I use it as backing. I have tried to use an iron on stabilizer but the minkee got too stiff. I would prefer to stabilize the t-shirts with a light weight iron on stabilizer. Minkee stretches and so do t-shirts. That could be trouble if you don't use stabilizer.