In my opinion..If it is just finrot..I don't believe you would have to tear the whole tank down.Carbon filtration is a personal preference, it is not 100% necessary. It does however help with odors and helps get rid of unwanted chemicals and impurities that may end up in the water. I use a piece of aquarium sponge stuffed into my filter to help hold good bacteria.If you decide to get a new filter cartridge just to be on the safe side..I would leave some of the old in your filter for a week or two so that it can help feed the beneficial bacteria.
However...I am not 100% sure how long the beneficial bacteria will live in your tank without a fish in it?...So you may go into a re-cycle anyways once he is added back..
How are you treating him for his finrot? I have had good luck using aquarium salt on minor cases and very good luck using maracyn 2 on severe cases of finrot. Hope your fish heals up nicely!