So the problem is the fuzz is coming to the top of the quilt?
I've only used minky a few times and I have a friend who has used it a lot. It hasn't been an issue but it was all long arm. Did have stretching problems once when my friend set up a crib quilt and then didn't quilt it for a couple of months, the minky sagged quite dreadfully but was still usable once I removed the partial stitching and turned it 90 degrees.
I know from garment/toy construction and plush fabrics that you have to prepare your seam allowance first and then go back over the seams by hand/brush and pull the fibers back out to make the seams nice/disappear. Those little manicure sticks are good for things like that, or maybe a toothbrush. Boring task but not so bad to do while watching tv or something.