You could make copper weapons for training, they aren't nearly as dangerous as iron, steel or obsidian weapons.
The tricky part is getting them to switch to training weapons. You could switch everyone to fighting unarmed. After they drop their weapons, have a mechanic build a huge weapon trap with every "good" weapon in the place. Once those are safely locked up in the trap, tell your soldiers to go back to using weapons, the only weapons around should be the copper ones, so that is what they will take. The problem comes if there is a military emergency, and you want everyone to quickly switch back to you good quality weapons. You'd need to switch everyone back to unarmed so they drop their copper training weapons, set a mechanic to disassemble the weapon trap freeing those weapons, and simultaneously have another mechanic build weapon trap(s) with all the copper weapons, to get them out of the way. Then tell your soldiers to go back to using weapons, and they should go pick up the recently released obsidian, steel and iron weapons, but it could take a while.
It would be handy if there were some way to designate certain weapons for training, and have the soldiers automatically switch to training weapons for sparring and then back to regular weapons when they are done sparring. If there are no sparring weapons available, they will spar with real weapons. If they are attacked while holding sparring weapons they will use those weapons to defend themselves, they aren't useless, just less dangerous than obsidian swords.
They could even be built and named differently, to make it easier to tell how many regular weapons and sparring weapons you have; a wooden training sword, a wooden training axe, a copper training mace, etc. Sparring weapons could be made from wood, copper and perhaps tin? A few choices of material would make it easier on maps that happen to be short on some materials.
Alternately, perhaps the clothworking or leather shops could make padded training covers for some weapons. That way you still train with your regular spear or favorite sword, you just put a padded cover over the pointy bit to prevent accidentally removing your buddy's spleen. They would still be dangerous, but perhaps only 50% as dangerous as a naked blade.