Is the flag valuable and not easily replaced?
Are there hems and seams that result in any "topography" to the flag?
Is there any hardware, like grommets, for attaching to a flag pole?
I would counsel against dry mounting anything of particular value. Despite Bainbridge's claims of reversibility there have been instances where that was not the case.
If it is valuable, sewing is probably one preferred way to go. I would suggest sewing to a cotton fabric that is stretched on a stretcher frame with a thread that's a little weaker than the fabric. Mat board and fabric have different dynamics when it comes to reacting to environmental changes, where fabric to like fabric doesn't.
The other would be to do a DCO (Direct Contact Overlay) as long as there isn't too much of the "topography" of fabric or hardware.
Here's a link to an ongoing discussion about a piece using the DCO technique.
https://www.thegrumble.com/threads/dco-silk-scarf.93163/ Dco silk scarf
If your interest is piqued, member Jim Miller literally wrote the book on DCO, and you can send a DM through this site to him about that.