I find it hard to believe that Hunter Arms Co. would pass the barrel in question with that thin of a wall in that area. It is also hard to judge how far back the thin wall is compared to the heaver wall down from it. At this point it would start to curve up but if the heavier wall was say .150, the thin area looks to be half of that and would definitely not pass their inspection.
Back then Hunter Arms proof tested their guns with twice the powder that was used back then. There were no proof marks stamped on the flats until the 1920's.
Also the groove that looks to be even with the extractor down from the breech would not be from the company.
Be interesting in what is found in the left barrel.

David