What I would do if I were in your position is to cut your batting in to strips - manageable quilting sized strips and start on the left side of your quilt, quilt the top, the batting and the backing. Don't quilt all the way up to the edge of the batting. When you are done with that section, fold back the backing and the top then push the next piece of batting along side of it and either whipstitch it by hand or do a very wide zigzag with your machine to join the batting pieces then quilt the next section. You won't have so much bulk to deal with because the batting won't be adding to it. Can you tell that I am not a hand quilter? No way, no how. I don't like doing it.
You can roll up the areas that are not being quilted and clip them with big hair clips or clothes hangers or something like that. The batting is the bulkiest part of it all.