I actually solved this before Mike showed us how to because I hate the stress of things not working.
Mike simply applies our mirror although I found a way around this.
Even though we childed the mirror axis to our “lower arm” mesh, for whatever reason, blender seems not to like this when we rotate our lower arm mesh with a bone.
Although you could change the way you are mirroring the arm, instead I found that if you parent the mirror axis directly to the bone, it will rotate it just fine.
^mirror axis parented to arm bone rotating just fine.
This way you will not have to redo your mirroring later when you want to add or modify the arm mesh.