According to this map on the Bristol City Council fishing page the corner of Baldwin Street and the High Street below the Bristol Bridge is OK for fishing.
As I could cast from the office window into the river (or is it a floating harbour?), I figured it'd be worth a look?
From looking at it, there's some steps down to the water level that'd be a good spot to start. Thinking a bit of pre-baiting campaign, gotta be worth a loaf of bread and a can of corn a week for a month?