I took apart the sink cabinet and fridge cabinet in my 71 a while back.
It looks like alot of work, but if you're systematic, than it is quite easy. And you are right, it is best to at least partially disassemble the cabinets to get at the tambour hardware.
In my case, I found added benefit in being able to clean thoroughly behind the cabinets, eliminating a major source of odor, as I found out after doing the repairs. Also found a very small drip from a water line under the sink.
Prepare for a small " can of worms" whenever you start disassembly of a vintage Airstream.