In order to level up armor/shield/helm, you have to equip and use an item while running around in the clockworks. Weapons are different in that you can carry a weapon in an extra slot, never use it, and level it up. I would suggest one of two things:
1) Divide heat between weapons based on how much you used that weapon throughout the last depth. If you're relying on your level 10 leviathan, then that wastes heat that could go to your other weapons.