If the center is waste, why couldn't you forego the collet and do it like how you made the collet, or do it how i showed previously, with pressure turning against the jaws?
If sticking with the collet arrangment, perhaps a wrap of thin tape might add some protection to the markings.
Very few things are waste in my shop. I deliberately trepanned it to save the center as stock for a future project.
Ya, I considered tape, but it wouldn't have saved me, the part didn't turn, it got gouged by the center hub that had turned into a sharp bladed turbine.
Next time, I'll go slower and also measure progress. When it gets close, I'll stop and then cut it out with a nibbler or a punch.