Poly on poly is slippery, so that's probably causing the shifting. I don't know that machine basting around the quilt will solve the problem for you, as the batting can still shift during basting.

I think I'd heavily safety pin baste near the edges, then put the binding on. Elmers washable glue might also work to stableize the edges.

I serged my last quilt that I just made and then put it in the washer prior to putting on the binding. It was very easy to to put on the binding, no bulk. The only time I did not sew around my whole quilt prior to binding I had an issue with a tear - when sewing a round the edge it also gives you a second glance if any seams are coming undone Bettr to fix it before the quilt is done.