Was thinking about this last night, after you quilt it for the first wash it would be worth it to go to a laundromat with one of the extra large machines. I do this for all my queen and larger tops.
After that you should be able to stuff it into a normal large capacity washer, but probably it will never fit in a stackable unit.
Consider what quiltingshorttimer said too, use a very thin batting (no warm and natural!). Maybe try just putting a piece of flannel inside, that gives enough layering to give the quilting stitches something to bite into.