Definitely baste to muslin to stabilize the top.
Since this will be a first washing of the top, I would use Synthrapol -- which suspends loose dye particles in the water so they can be rinsed away instead of settling into other fabrics. If the red bleeds a lot, the Synthrapol will definitely help even if you do not use hot water. Be prepared to run a ***lot*** of water into the tub and let it run out again; it may take quite a few changes of water if the red is running.
Aside from the problem with strings ravelling from all the seams, I think the biggest issue could be unequal shrinking of the different fabrics. Basting to muslin first will help with this. I would also be prepared ahead of time to block the top when it comes out of the water. Blocking before it dries would help the top retain its shape and prevent fabrics from shrinking too much.
If at all possible, it would be much better to quilt the top first, and then wash the entire quilt. This would eliminate the ravelling problem and the shrinking problem, plus you could wash the finished quilt in a washing machine with no problem (much easier in my mind than bending over a bathtub, especially if the red bleeds).