Have you tried blue tape and CA glue? It would seem that the blue tape would hold the material but be easily removed after being cut. I have seen many videos on YouTube with wood and other more solid materials being held quite well, but never thin materials like you are using. Seems like it should work though.
Let me know how it works. I have not had the need to do thin materials yet. I know it works on the stiffer thick materials. Also, if Blue tape does not hold well enough try Orange tape. IF it sticks too much try another color. I didn’t know the colors meant something until my son, a commercial painter, told me the different uses for different colors. Orange is more sticky than blue.
Just by way of follow up…
I bought some of this tape, this brand specifically on recommendation of a CNC youtuber. So far it has not let go of a project and has peeled of well after the cut. Bits do not like it if it is in the toolpath on a cut through though. I am not sure what it would do on the thinner materials you were using.