Rip the backings around 3/16" thick. What sort of saw are you going to use? With a table saw you have to be mindful about the backing falling between the blade and the saw plate if it doesn't have a zero clearance insert. Saw on the outside of the blade(not between the blade and the fence where the thin backing can get trapped. Use a push stick. Or you could use a bandsaw, which is safer but a bit more difficult to tune to get an even cut.
I would sand both saw surfaces to get rid of any burn marks(if you didn't up a rip blade),80 up to 120grit.
Glue you could use Titebond 3 or a good epoxy.