I use a lot of minky and generally do use polyester thread, it is good and strong and Minky is polyester. I would use a stretch stitch ( tiny spacing zigzag and an overlock could be beneficial for your project. I would probably put the filling into a muslin enclosed pouch first then put it into the Minky case just to keep anything from migrating out. If I have to cut Minky I always put painters tape along the seam im cutting which keeps all the fluff from getting on everything. I use 2” wide tape and cut down the center of it. I know you have already made your cut, but tape along the cut edges will keep the mess to a minimum.