In viking society it was actually required for all grown men to carry some sort of weapon with them. I think this was considered proof of free status, but that might have changed from place to place and time to time.
This has probably been in one of the previous topics, but it makes sense to me that you should be able to define types of weapons as civilian weapons and others as military weapons, so i.e. staff, club, seax, hatchet are civilian, sword, battle axe, spear, etc, are military. Or it could be based on materials, civilians get wood, bone, stone and bronze while military gets the steel and adamantine.
If civilians could carry weapons, it would be great to have some sort of meeting area that works like a gym or dojo or arena where they can practice martial arts, when they feel the need to. Dwarves could spar with each other or share knowledge about martial arts like the scholars do (I guess that could be the same as the squad demonstrations). Crippled militia members could be assigned to the dojo to pass on their knowledge to the next generation. Might be possible to have duels or tournaments or something pop up from time to time as well. Combat treatises were popular in Europe and Asia, though perhaps not as much pre-1400, maybe there could be a way to wright manuals on martial arts skills.