Easter both the cd for the look, the backer for a press fit or even just a down set so you know perfect center placement and then cut of both objects and both materials at the same time
I wonder if glowforge ( or regular laser cutters) can make the image on lightscribe CD/DVDs. I still have quite a few of those. Remember - if you had a lightscribe drive in your PC you flip the CD upside down and the cd laser would engrave any image or wording on the backside of the CD. It even did photos!! Not sure of the power of a CD drive’s laser, but seems you could dial-down the power of the laser cutter and 'scribe the back of one of those with an image or text, etc.
I still have a few CDs left as well as an external lightscibe dvd player, so I can use it with current laptops etc ( assuming I still can find drivers/sotware)
After my Lightscribe burner went belly up, I tried to find another one…to no avail. Then, I bought a new iMac and the software for using Lightscribe would no longer function, anyway. After a small period of mourning, I have moved on.
After my first experience with a label coming off a CDR while spinning in a slot-drive, I moved to sharpies.
These days I like the Sakura Permapaque for shiny surface marking.