If you want to use minky, I have been told to use an iron on stablizer to keep it from stretching

How about strips of fleece (cut with pinking shears) to make the binding from, still soft and looks intentional?

You might post a picture of some of the fabric used. Many times I have seen quilters on this board come up with the exact fabric to match that they would part with. Worth a try.

Minky will stretch an slide. Why not just use a coordinating fabric for the binding. Pick something that blends with the ret of the quilt.