I just logged in to see the situation for myself and am able to start the unlock chain even though I did the Mechagonian Threat on my Horde main. The flag in the embassy shows only the rep done and not the achieve but I was still offered Urgent Care which is the first quest in the unlock chain. This is on an Alliance toon that is also Exalted Mecha even though it has not completed the dungeon.
Not sure if that is a recent change they made, nor have I completed the chain yet to see if it actually unlocks.
 Finished chain, unlock done. So it looks like having an Exalted Rustbolt toon Alliance side was enough even though the flag was telling me I needed the Threat achieve too.