I imagine cloth would work the same as paper and light cardboard’s.
Get some ‘Krylon Easy Tack’ (or equivalent) from Amazon or a local hobby store.
Take a cutting board board or a slab of plywood to make a tack board with. Less than 12x20 will allow you to nail it down to the crumb tray with magnets or pins (just make sure it is larger than material you will be working with).
Measure the board thickness. You will add this to the thickness of the material cut, and then continue to use the values you have found that cut the cloth nicely.
So if board is 0.180 and material is 0.020, then keep the cut, engrave, score values you have been using the same and just change thickness to 0.200. You have essentially raised the crumb tray by 0.180.
Spray the board with a light coating of the Krylon (will be good for several cuts sessions) and place into the glowforge. Flatten the material on the board (use magnets or pins on excess material to keep it from blowing around).
Cut, score etc. Replace materials for additional cuts. The cut sections should stay put and can be lifted and pulled off the tack when done.
Lot cheaper then a purchased mat, and can be easily replaced if mess up and chop it.