What if...you use 2 layers of white (or muslin for the under layer) for the sashing and then trim the colored fabric in the seams to slightly less than 1/4"? Maybe even do slightly larger seams before trimming. Another option might be to zig-zag the seams or use a serger on them if you have one.
I admit I've not done this but the latest hand embroidery suggested putting muslin behind the piece before doing the hand work and it works.
Let us know what you decide to do and post a picture .