Maybe a double mat with mounting corners on the inside? This way you can open the case and swap out photos very easily. You can have one double mat for two 8x10s and one for the larger size photo. The overall mat dimensions can be the dimensions of the case, and you can simply use linen tape (or even something stronger) on the back of the mat to the inside of the case. Honestly you can even thumbtack the inside mat to the corkboard, the tacks shouldn't be visible through the outside mat.

I've been using Frame Destination for my custom mats and frames, their quality is outstanding and their prices are reasonable.