Check the retorque of the gas block. If the two sides of the gas block are un even do the retorque and keep in mind to keep the gas bushing centered. Since you can't see the gas bushing it's done by "feel". It can be done.
I like to smooth up my barrels with Brasso. Yes, the stuff you used when you were in the Army to clean your brass. There is also J B compound that is used to smooth up the barrel as well. I prefer Brasso. Remington makes a barrel cleaner that is similar to Brasso as well. I think it's called 40X or something like that. Brasso has what ever is in it that gives it it's distinctive aroma and seems to dissolve carbon and copper. Just use a nylon brush.
I shoot Tula out of my AR's and Mini's. It's a plinking round and will not get you 2" at 100. It may get you 4 inches at 100. I can't tell you about Brown Bear, I have never shot it.
I have used aluminum tape to shim up a Mini receiver in a plastic stock. It seems to work. Nothing beats glass bedding a wood stock.
I have a 581 Mini in .223 and I really like the factory sights. I'm reluctant to say there is ANYTHING better, but there may be.
My opinions. YMMV. kwg