You might look into acid free archival matting paper for your project. It comes in lots of colors, and I believe it comes in various thicknesses which you could choose from to get the look that you want. You’d need to research specific products, but it seems to me that it should be safe for coins. Amazon has a good selection. A cutter like this one might give you the adjustability that you need for a good fit, though I have never used this product.

Fiskars 193800-1001 Circle Cutter with 2 Replacement Blades https://www.amazon.com/dp/B0006HUJ0S/ref=cm_sw_r_cp_api_glc_fabc_2j87FbZNKPAX9

Just suggestions on where to start.