Bridewell looks boss (and run by Dom and Fiona--good folks who I met when Dom was at Fly in the Loaf). But it wasn't there when I was staying in Liverpool back in 2014/15.
Back then we were staying at University of Liverpool, so we tended to go to Fly in the Loaf, Caledonia, Belvedere, Philharmonic, and Grapes (sometimes the Blackburne for brunch). Other places I liked were Ship and Mitre (very genuine place), Berry and Rye (for a high-priced yet fun outing), Baltic Fleet (another genuine sense of place here), Roscoe head (when you wanted to feel old man-ey) and Pen Factory (was great for a loud open-feel stay w/food...some good German beers if I recall).
Damn, miss your excellent city!! Hoping to return soon. ( Edit: I'm pretty sure I say all this every 2 years or so)