I have used two methods for holding down PCBs to cut that may work for you. PCBs are somewhat thin and need to be held down everywhere very firmly to prevent high and low spots, somewhat similar to your needs.
As mentioned above, there is the masking tape and superglue (CA) method. I used blue painters tape, and it worked great. I think this is probably your best bet.
A little easier is this double-sided tape, which isn’t too sticky and removes very nicely: https://www.amazon.com/Nitto-Permacel-Double-Coated-Kraft/dp/B000QDL9GK