This is what I would do.
I would dampen the patchwork before putting the borders on. Get it rather wet but not in the washing machine. Soak it in water and squish it dry. Lay it out and stretch it to dry and when still damp then iron it with hot iron.
Then square it up.
Prewash the fabric for borders and backing and Iron and then put together.
I would definitely put a very thin batting inside. Perhaps something like Hobbs Thermore. I believe it needs some stability in the center. Thermore or other very thin batting wont add much weight but will fill it out somewhat.
Another layer of flannel would be too heavy for me.
I believe this will prevent some problems in the future in shrinkage since it most likely will be used, lovenly abused and washed often.