There is a Brewer’s Arms on Marygate, and that pub is there, in the same place, on the earliest OS map that I have seen at NLS.
When I search for ‘Brewer’s Arms’ at BNA, and limit Place to Berwick, the earliest hit is in 1859 and refers to ‘Brewer’s Arms, High Street’ as do many subsequent entries.
Conversely, when I search for ‘Brewer’s Arms Mary Gate’ the earliest hit is 1922.