In the 1855-7 Old Town Survey Map I have found a P.H. (Public House) marked behind the Montpellier Baths (Playhouse theatre) in the far north-west corner, accessible from the Baths and by an alleyway which runs along the back of them then turns 90 degrees north to link with Wellington Street. I am guessing it probably closed after disastrous mid-1850s floods saw parts of Bath Road around the Baths under up to 5ft of water.
Information courtesy Alan Jones