THIS/\
I've taken several dents out of shotgun barrels. Even went so far as to build a hydraulic expander, made out of rolled brass around a heavy hydraulic hose, attached to a small hydraulic jack used for multi tools.
It worked, in one case, to well. Good thing it was a well rusted and damaged practice piece.
You need to be especially careful around side by side or multi barrel combinations. It doesn't take much to separate the barrels.
Another thing you need to be extremely careful with is the edges of the mandrel you make up. ROUND THEM OFF. SHARP EDGES WILL BE ETCHED INTO THE BORE. I've seen this several times.
READ GUNTECH'S words. LIGHT TAPPING is the key. I use a piece of aluminum/titanium as a mandrel/anvil.
You will very likely need three hands to do this job, if you don't have a decent vice, with SOFT JAWS to hold the barrel properly. The barrels should be removed from the action or all sorts of other damage can and will occur. I also use a SMALL BRASS TIPPED HAMMER. It's soft and won't mar the finish. It will however leave some light residue behind, which cleans off easily.
YOUR BIGGEST TOOL will be patience and common sense. OP, your shotgun muzzle is an easy fix, but could go south in a hurry if you get ham fisted.