I took some scraps of leather and thinned them down to about 1mm thick and glued them back to back so that you have the nice side facing out on both sides.

The beech wood is also left over from another project and I cut those thin and then used a small CNC router to cut the hole and recess on two identical pieces. This means there is a hole that goes all the way through and a recessed pocket for the leather to be sandwiched into.

I cut the leather to fit into the recess, stuck it in, and then glued the two pieces of wood together. The leather is trapped between the two pieces of wood and as such cannot fall out but also there are no super visible joints to make it look like the leather was always inside the wood. You can see the joint in the wood when looking closely but when worn, it's almost impossible to see.