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.
2) If we want people to be able to level weapons up without using them, allow it for other items too. An extra slot could be bought that could hold any item. You would not be able to use this item, but it would collect heat. I'm not a fan of this, but I'd rather have this than a special privilege for weapons.
I like this. I personally support option 1, although giving the heat to your dresses would be also cool as an alternative to 2. And enable shield-dressing, while we are at it.