It is amazing just how little you actually need even for cooking and camping. A credit card tour requires almost nothing other than whatever tools and spares you carry on the bike around town. Personally, for a credit card tour, I'd probably ride a road bike with no racks (if you own one). A large-ish handlebar bag or seat bag would be plenty of space IMO. Heck you could carry everything you really need in jersey pockets.

Whatever you do... Have a great trip.