So just to experiment, I cut a lantern sized slide mount for 6x7 (Linhof 56x72mm format) from 4-ply mat board using an X-acto knife and a straightedge, and other than my cuts not being perfect, it's not a bad thing!
I mounted an unimportant transparency and projected it for a few minutes, and the gate actually stays cooler than I thought. The slide didn't even pop, though that may have to do with my mounting technique, which was to tape it on one side only, so the slide can expand in the mount.
The film isn't quite as flat as with a glass mount, obviously, but a 6x7 slide with this wonky setup still looks WAY better than a 35mm slide.
Maybe I'll just order a few batches of mounts from one of the companies like Redimat that sells custom cut mats, or I'll finally buy myself a decent mat cutter with production stops. I've used a Dexter mat cutter and a T-square for years, but for these little mounts, there's not enough mat to hold on to that way.