I guess it’s stating the obvious, but if you camp more than once in a blue moon, then it might be worth investing in a comfy self-inflating mat for car camping. Alpkit’s Dozer is 64 quid currently and big, 7.5cm thick unit that’s very comfortable, but also packs down easily. They do a double version tool.
Otherwise, straps. Or possibly a large-ish compression tent bag – MSR do them, tho maybe not large enough, but at that point, you might as well put the money you’d spend towards an actual self-inflating mat.
OR you could stuff it into an appropriate cardboard box and then tape it up. Or some sort of plastic barrel container.