They should work fine together. However, you really need to wash and machine dry the flannel twice to let it shrink before using it. Flannel can shrink a LOT more than quilting cottons, so it's a good idea to let it do what it needs to do before you start cutting it. The shrinkage tightens the weave too, which is a good thing.
After washing and drying for shrinkage, I highly recommend that you starch the flannel yardage heavily. I use a 1:1 solution of Sta-Flo liquid laundry starch and water applied with a large wall painting brush on yardage, allow a couple of minutes for the fibers to absorb the starch, toss in dryer, then iron with steam. This stiffens the flannel and stabilizes it so it won't wiggle and stretch while you sew it to the cotton. Only after starching would I fussy-cut the dog faces.
It is not necessary to pre-wash the quilting cottons unless you like to do that. The amount they shrink is very small compared to how much flannel can shrink. Assuming you do a moderate amount of quilting (lines at least 3" or 4" apart), everything will hold together just fine.
I think your idea of soft doggie faces in frames is terrific!