I would think card stock is nowhere near stiff enough to wrap fabric around. It's only the thickness of a greeting card.
Foam core would work and comes archival, but it's not cheap and can be hard to cut. Mat board comes archival (you have to ask for it) and is easy to cut, but it's quite expensive.
The comic book boards that everyone is so fond of are only archival on one side so wrapping fabric around them is really the same as non-archival cardboard.
I'd say the least expensive and most practical way is to fold the fabric with nothing inside that you just have to remove anyway when you want to cut it.