I've incorporated some non-ideal fabric into quilts before... but yeah, this is a challenge!
I'm thinking that this is the label or maybe center square of the project, I imagine they have some embroidered crests for their department somewhere like shoulder patches? Or if you/someone you know has a machine that can digitize an image and embroider the unit name and pertinent label info.
The other thing I can think of is maybe make a tote to place the final quilt in using the jacket/trim in that.