I also wouldn’t have guessed that refinishing could harm the beautiful inlay work. No marking on the barrel at all.
Refinishing
per se will not harm the inlay work, as long as it is done correctly. The best finish is of course aqua fortis stain + a non-polyurethane oil varnish. You can make aqua fortis from iron nitrate crystals in alcohol, or if you use a nitric acid / water base aqua fortis, just neutralize it with baking soda or ammonia. Skip the tannic acid. It's a fad, and worse yet, it's water based.
Adding water to the stock could conceivably raise the grain enough to move the inlays. Maybe not. Or maybe they would be fine after the stock dried. Why find out?
So, KISS principal applies - I'd highly recommend alcohol / iron nitrate crystals. This will not raise the grain, gets ideal results and makes your life a lot easier.