OP- even if you convince Mom your child does not need to have gun and that the other two boys should not have shotguns, there will probably still be guns around due to hunting, Mom's boyfriend, etc. Mom may even take up hunting, or own guns in her home. So, it looks like the likelihood of guns being in and around the home will be present.
I completely understand your concerns with your son, but I also recognize there's a strong possibility that if he is the only child in the home without a gun, guns may also become more attractive to him as something "forbidden" is often more interesting.
What I would suggest is if you and Mom can't come to some sort of understanding after discussing the issue (which doesn't appear likely), I might consider purchasing trigger locks for all 3 children myself, and maybe even cases (soft cases are not as expensive as hard cases) with keyed padlocks as a Christmas gift from you. Ammo boxes are also very inexpensive (just another keyed padlock will work) and, in my opinion, ammo stored seperately from guns and under lock is another very important safety factor. Doing so, maybe you can voice your concerns to Mom and she will have a more open mind, and you will be helping to ensure safety at the same time.
Is your area an area where lots of people hunt and/or own firearms? If so, a local attorney could probably tell you what the Court climate is like regarding firearms, particularly shotguns.