I shoot factory loads rarely and mainly low brass target 12 gauge. A lot of Gun Club and a lot of Winchester AA. It does seem to me that the AAs fit a little snugger than do the Gun Clubs, but I question that the slight difference in the OD of the brass is the problem.
It may just be that polishing isn't enough and that the chamber of this gun is just a bit tight. I had a buddy with a new 870 Wingmaster that was so tight it needed to be reamed out by a smith.
The issue is whether the chamber is so tight to interfere with extraction. I believe that Mossbergs have twin extractors or is that just the military model? Anyway have you checked them for wear?.
I would think that this is a problem that a smith could evaluate in minutes. It just may be that a tight chamber resulted in damaging the extractor and that it needs to be ream out and have the extractor(s) replaced.
Try this. Using new ammo, check it for any scratches or burs. There should be none. With the barrel removed, push it in with your finger as you described. Do so with a reference point on the head, such as the W in Winchester at the 12 o'clock position. Do not twist it when pushing in or out. Push it out and then carefully examine the brass for any marks that may have been made by a bur. Do this with a few shells. If they all have a similar scratch in the same place on the clock you should have an idea where to look for a bur.