Have you ever basted on a garage wall?? Yep, in the garage! It's a great place to spray baste with the backing hanging full open. and the door up to ventilate.
But in your case it could also serve as a place to layer and batt a quilt with fusible batting. I can sort of see stretching the backing or top onto an 8ft x 8ft. (use 2 @ 8'x4' from Lowes) foam board, upright position, and pinning it down before pressing the batting down -- with your HANDS -- as a way to straighten it. Then proceed from there.
I used to use Hobbs fusible batting all the time....wonder how I got away from that??? Good luck!
Jan in VA