You'll do far more damage to the bore by not cleaning it properly than any bronze or synthetic brush ever could.
I recommend buying a good bore solvent that is made to help remove plastic fouling caused by modern plastic shot shells and plastic sabots.
Buy several bronze or synthetic shotgun brushes.
Use a hobby paint transfer bulb to apply solvent to the brushes and patches so as to not contaminate the solvent.
Apply a good amount to the brush , and push it ALL THE WAY through and OUT THE END, before pulling it back through. (NEVER reverse ANY bore brush inside the barrel).
Brush until the fouling deposits are removed, (At least 15 to 25 passes of the brush), keeping plenty of solvent in the bore.
Then run several soaking patches down the bore and out the end.
Allow the bore to soak while you clean the rest of the gun.
All bore solvents need TIME to work, so allow plenty of soak time.
After soaking, run a couple of clean patches through, then using a good light, take a close look at the bore and chamber for any signs of fouling.
If you see any, go back to the brush and solvent and scrub some more until the barrel is truly clean.
Then dry with several patches, apply a good rust preventing lube like CLP Breakfree to a clean patch and coat the bore with lube.
Allow a few minutes for the lube to spread, then use another clean, dry patch to wipe out the excess.
Many shotgun shooters also buy a special chamber cleaning brush and handle to insure all plastic deposits are removed from the chamber.
The major cause of extraction problems in shotguns is a fouled chamber caused by not really getting it clean.
Bore brushes often will not do a good job in the larger chamber, and shotgunners are often shocked to find their chamber is pitted under a layer of plastic fouling they failed to get out.