Closed between 1884 and 1902?
38 Bridewell St
grid reference TL 770 458
Shown as an un-named beerhouse on the 1884 OS map. It seems to be the building now known as "The Old Bakehouse" at number 38.
It doesn't appear on the map surveyed in 1902, so presumably had closed by then.
The beerhouse is visible on this old OS map from 1884 (interactive map). The pub to the south of this was the Old Red Lion.
Old OS map reproduced with the permission of the National Library of Scotland.