If a 20 min walk’s ok, I can recommend some nearer to Borough:
The Gladstone Arms,
The Lord Clyde.
The Lord Clyde and the Gladstone are small and more traditional and usually get missed as they’re a bit off the track. The Libertine is a massive favourite, usually for their pizza but their beer is good and they now have Augustiner Hells on bottle. The Roebuck is nearer to where you’re going and is just a nice place.