I agree that you should be fine. I have several with a bead blasted finsih and have never had a problem. My son has a couple of the Browning A-Bolts as does a friend of mine. Both of theirs have started to rust and pit. My sons wasn't too bad and we were able to clean it right up, but my friends finish was just about ruined. It was his own fault, he left it in a closed case. These were both Brownings and I'm convinced there is something different about the composition or type of SS that they use.
What we started using on my sons is a rifle/gun protective wax made by Flitz. This stuff works real well and we only apply it once a year, usually before our first fall hunting trip. I started putting it on mine as well since I was so impressed with it. It's similiar to this polish but I think it's marked gun-rifle wax.
FLITZ : FLITZ LIQUID POLISH - World's Largest Supplier of Firearm Accessories, Gun Parts and Gunsmithing Tools - BROWNELLS