You might take a look at the K40 laser cutters. They are cheap to get and dirt simple to operate.
I bought one on sale for $300 shipped a couple years ago and had it running in under an hour. I liked using Inkscape for the input files because I could stick a photo in it, have it trace the lines, then cut (or do manual cleanup if I wanted).
I have not made gaskets on it, but I cut 4mm plywood and acrylic regularly. I would not be the least bit concerned about it functioning.
EDIT:
There is a lot of good information on it here:
https://k40lasercutter.com/I loaned mine to a friend for a couple projects. If someone has a couple scraps of the thicker gasket material they are willing to donate, I could do some test cuts and report back. Or if you had a gasket you wanted made, send me a pic with a couple dimensions on it (so I can ensure the scaling) and I will make them as a demo.