I found the solution after spending a few hours retracing the first person setup from the template and considering what you said. What fixed it for me was that I had to move the weapon_mount bone, which was at world zero, to the grip bone of my gun for the animation sequence and it exported correctly. I have not checked if that will work once the weapon and all of its parts with be animated, but I guess that will be the case.
I guess I have to export the gun at world zero for the reference pose, then simply move the weapon_mount with the weapon grip when posing.
Spent the weekend figuring this workflow out. Your posts have been a great help.
Feels great having this work.
(The gun and hands are marketplace assets, but I rigged it myself, and the animations are from scratch too)