A loadbearing vest ought to be able to be flipped inside out and stitched in.
I don't see why a good tailor couldn't add armor panels to an already thick jacket. The only problem area being the front closure. It would have to be more like a button down style with the big overlap. Which would also be largely negated by having to open the jacket to access anything. Granted, I'd rather take my chances with armor that may not be in place, than no armor at all.
I'd worry about the armor rating, since it usually requires it to be snug to the body. You might just be better off with a concealable vest under your shirt, and then whatever else.
Fwiw, walking around, I can pretty easily carry two double stack 45 mags, an AR mag, and said 45 pistol on my belt, under the jacket.
Sometimes I lounge with a 380 in the breast pocket of my zip-up hoodie. It doesn't take much.
If I remember correctly, Carhartt or Berne offered a ccw jacket and vest which used a velcro+snap outer pocket to hold a gun. I don't think it's the greatest idea nor execution, but I'm sure it works for some. Maybe I should rethink it, since belt carry tends to get in the way of my fannypack-manpurse that I use when small game hunting to hold shells. Or on a super cold day deer hunting, where my sidearm must be concealed (stupid Ohio game law). Usually those days I will put my pistol condition 3 in the breast pocket of my bibbs (a kydex trigger guard thing could remedy needing condition 3, or not carry a glock)