I could be wrong here -- haven't had a chance to test this -- but I pickpocketed a key from Arkur during my first encounter with him. After he got the call on the intercom and started to leave the office, I snagged what IIRC was called Arkur's Key. Regardless of the name, however, I do know for certain that at no point in the rest of the DLC (nor anywhere within Haven) was that key required to open anything, and I explored pretty much everything there was to explore.
So -- key to the front door, or key to the executive bathroom? Not sure; though I will try to post back later tonight after I get my first chance at some post-Pitt exploration. In the meantime, I hope that helped-ish-kinda? :)