i hope you thoroughly re-oiled ENTIRE gun to prevent rust.
How about under the stock? did you clean that area too?
brake cleaner is harsh on grease (and your health) it penetrates fairly deep into the top lair of steel/bluing and devoids it from protective dried up oil residue.
if you use breakcleaner you must re-oil your gun very well or you will see rust froming quickly when humid weather returns.
there are many ways to clean cosmo. i personally prefer elbow grease with generous help of wd-40 which softens cosmo to Vaseline-like consistency that's easily wiped off. this method preserves dry oil film in the metal and prevents rust.
anyhow there is no need to scrub down entire rifle, inly places that really need good cleaning are action, bore and whatever area you will touch and don't want cosmo to stick to your fingers. certainly no need to clean under the stock, i'd even recommend against it.