Hello, all,
Brake cleaner is a degreaser, not a penetrating oil or a rust dissolver. It's not appropriate in this case.
Do not take the barrel out of the action, sir.
I would use the Break-Free CLP or one of the Liquid Wrench aerosols. Spray the area liberally while the barrel'd action is locked in a padded vise(wooden blocks would be OK), spray down a rag heavily with the same penetrant and push it into the area that is corroded. Leave it sit for a couple hours. Scrub it out with some SCOTCH BRIGHT, HEAVY DUTY SCOUR PADS BY 3-M, like you use on pots/pans/dishes.....so you are not leaving broken pieces of steel wool in the action to eventually rust out themselves and the metal they adhere to. Those Scoth Bright pads will remove gun bluing, and they'll actually polish stainless steel --So Be Careful where you rub.
Repeat this process a couple times using a proper penetrant and the Scoth Bright pads and you should knock down all the rust.
Clean all metal with the penetrant on a decent rag, rub some kind of heavy duty EP grease containing moly' into the action and the barrel, etc.etc. to stop rusting (to a very good degree....though not salt-water proof!). Wipe off the excess grease after a couple days if you have the time.....but go hunting with it in any case -- once the chamber and bore have been thoroughly degreased and dried out with Brake Cleaner.
Good luck.
