Frame challenge - Leave it at home
Like you said, folding a vinyl mat is eventually going to damage it and even if you roll it carefully around other stuff there is no guarantee it will stay this way as plane luggage tends to be manhandled quite a lot, plus it's an extra weight that you're going to have to carry around, it's just not worth it for a single short session that you're just considering running.
Use a paper sheet instead - you can either glue/tape several standard paper sheets together or buy a big poster-sized one if you need to, you should be able to get either easily at your destination unless it's somewhere really exotic. Or use a virtual table top.
If you really have the need to take it with you, the way I would do it is by first rolling it around some flexible tube (like a foam noodle), then arranging it very gently along the edges of the suitcase and packing it tightly with clothes or something else relatively soft so that it doesn't move. You're looking to round it rather than bend. That would be a last resort option for me though, I'd much rather leave it safe at home and improvise something than risk breaking it.