Have you heard of the envelope style back? It is similar to the birthing method, but instead of 1 piece for the backing, you have 2 over-sized halves (hemmed on one edge). Then you can turn through that opening, and do not have to hand sew the closing. You can fuse it shut if you want. Here is what I am talking about - eventho this is a pillow back - just use this for mug rugs. This is how a lot of machine embroidery in-the-hoop item are finished.
https://www.thewonderforest.com/2013...-tutorial.html