You might try a single-flute, or "O-flute" bit, which is optimized for plastic. https://www.amanatool.com/51515-k-so...outer-bit.html
With material that thin, PC-board bits might work too, and they're a lot cheaper (which is good if you're breaking them rather than wearing them out). Look at the Kyocera brand for instance.
To hold your parts down, you might look at a vacuum pod, instead of the glue and whatever. If you can get it held firmly to the spoilboard, you can use an up-cut bit which helps keep chips out of the cut and might help with the breakage problem.