I have found that on our laser, with “thick” material (this applies to things that smoke/burr like leather, as the char residue causes problems) it is indeed possible to get through >3mm with careful attention to the bed zeroing and brushing/scraping out the residue crumbs between 2 or more runs.
I have long thought about this but not tried it, as often the cuts get >50% through the material before causing issues - a potential method to overcome this is to make a simple frame jig with locating pins so that you may accurately cut from both sides of the material, much like how I have handled double-sided engraving on CNC machines in the past. I’m pretty confident you could get this to line up nicely enough for the cutting lines to match up. Considering that the rubber is like leather, you will get a zone of cut rather than a thin line, which would probably be quite forgiving for tiny mismatches in orientation.
As for the unknown aspect of the material and not being on the ‘approved’ list, we would have to have access to a MSDS document for it and some test pieces before we consider cutting in the space.