But alas, it’s all based on a false premise that I’ve left uncountered for far too long.
Let’s rectify that, shall we?
At the moment Genji walked into the kitchen to begin helping Gohda, the key to the gun room was still on him (on Genji’s person, naturally). The key had, up to that point, not once left his possession (which is to say, not once left his person). Of course, at the moment directly after Kanon left the kitchen, the key was still in Genji’s possession - which is to say, on his person. Finally, since you could claim Gohda pickpocketed the key, went and somehow got himself the gun before leaving, had Rudolf be outside, and then shot him from a remote place or something of the sort: Gohda did not pickpocket the key to the gun room off of Genji. Finally, to make sure it’s nice and sealed: Genji did not accidentally lose or drop the key or gave it to someone and then forgot about it, or anything of the sort that might involve the key leaving his person of his own accord. This applies for the entire game.