OP, which carpet do you have? Was it a specific brand, or a scrap of something?
I have a CGear sand-free mat, probably the medium size, so 8' x 8', but I never take it anywhere because, in plain English, it's such a pain the a$$ to fold and store. It's springy and slippery and uncooperative and I end up concluding much of the time that it's not worth the hassle. Just now, I tried to roll it to put it in one of my PVC tube scraps to see if it would fit, and it ain't happening. Or it could happen, but the effort makes it not worthwhile.
An alternative to a tube would be a tarp envelope into which a mat or carpet is folded and then stored under the chassis. I developed
this Valterra stacker under-chassis storage system (blog post). The stackers themselves are encased in tarp envelopes (gray packages in the middle of the pic below, secured to the frame with bungee cords) because the "lego" style pieces that snap together would not do well if exposed to a lot of road grit.
My CGear is easier to fold than roll, so I might make a tarp envelope to see how that works out. In a similar vein, I made a tarp envelope for our two UCO BBQs (the Flatpack and the Grilliput). That looks like this (blog post
here):
The Sprinter chassis has numerous open ports and holes to which either a tube or an envelope might be securely attached. If I manage to get my project successfully executed, I'll post back a description.