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Maybe a fabric paint would be closer, or a plastic paint. But who wants to be the test subject!
The recommendation of gluing vinyl is the best one I think. Should be easy to find a paint you would think, but new material will also look cool, protect the cardboard, and let you choose any color you want (as long as it's black?) too.
Or is the cardboard pretty deteriorated with cracks and missing pieces? That's the only thing I can think of that would preclude the vinyl solution.
I have used oil based paint on old interior cardboard before with good results... a lot nicer than "before" with no apparent swelling or other issues. I think Rustoleum or in general alkyd enamel would work fine. Maybe do a small test strip first to be sure (after cleaning as well as possible).
Don't laugh but shoe polish will do the trick, to be more specific liquid leather dye the old school not environmentally friendly stuff yes it is still available. You could either brush,roller, or spray it on.
Adam, I do restorations ( and used to teach ) and paint all the time. I can answer any of your painting questions. You have had some good info, so whatever you think you don't understand, the who or how, just shoot your question here.