This is pure speculation, but may assist.
Alexander Berwick and Co owned a brewery in Gentles Close, Canongate in 1825. In fact the Berwick family seem to have been involved in brewing in a number of places. Alexander was quite prosperous. Perhaps his father was the original brewer. It is possible that he is the Bailie Berwick referred to.
Gentles Close dids not have an entry onto the Canongate, but instead onto Back of Canongate - now Holyrood Road. It was just south-west of Bull's Close where there was a brewery marked through most of the 19th century. It seems to have expanded over time to become the Holyrood Brewery and was demolished in 1995. Gentles Close would now seem to be Gentiles Entry.
I wonder if this is the close that was referred to, before it had a settled name? Or that was how it was locally known?
Nell