I would start with stainless wheel on exterior parts. Polish bore with crocus and graduate to valve grinding compound on shaft with drill. Internals walls of action use Dremil. Give bolt assembly and trigger group a dunk in Kroil
then put them to the stainless wheel. The A5 has recoil spring & buffer, you may be able to clean them up, that steel is harder than rest of gun and may clean up easy. You can tell when gun is back together if you need new recoil spring. Other than that there is just the forked flat spring for trigger group. Usually friction rings will be ok, cleaned up.
I wouldn’t hesitate to put gun back as shooter. Since it’s only value is a shooter there is no point spending money on some kind of finish. I would get some good cold blue and have at it. I look at it this way. You are going to end up with $250 gun, my way. Having some finish outsourced you are going to have a $300 gun minus money spent on refinish.
I have done several shot guns including A5 & Rem 11. Never had one evenly coated with fine rust. The receivers on Brownings must be similar to older Winchesters. Some cold blues won’t touch them.
I would probably use boiled linseed on wood. Rub in several thin coats.it will bring water damaged wood back to life.
Done several and best way to deal them off is to prune them to 18” and rebead. I’ve spent hours doing above process on shotguns. Take them to shows and nobody even looks at them. Whack that dude to 18” and the first flite in the door will buy it.