My method would be to
hoop the main fabric
place batting beneath and stitch the trapunto area
trim excess batting
place more batting (&backing if you like) under and finish stitching
I don't use any stabilizer with batting.
This way you don't have to remove anything from the hoop till done. you may prefer to use a little adhesive to hold the battings in place, but I generally find it all stays put once stitching has begun.