I'm a big fan of scooterworks, but its the other side of SE1 to me - looking for somewhere nearer the london bridge/tower bridge area. Platform is somewhere I've been before but they tend to have very loud music. The Rose sounds a good bet, hopefully they still have the late license when they reopen.
The Victoria on Pages Walk (one street east of TBR) has a late license on the weekends. Fridays are usually busier and the volume unpredictable -- usually pop/disco stuff. Saturdays it's usually low volume and more like easy listening. When it's warm you can sit outside if there's too much LOUD in the Girls Aloud. You can never tell when you walk down the road what kind of night it's going to be, busy or not, boisterous or quiet, but it certainly fits the bill of not resembling a nightclub in any way. That's where I always go even though there's another old pub right behind my flat but then I prefer that sort of old-timey pub to refurbished fancypants ones. It might be worth a shot when you don't want to stray too far from TBR. You can usually get bank seating when you want to huddle up with friends.
I can really recommend the Thomas a Becket pub on the corner of Albany Road / Old Kent Road SE1 5UE. It's a great venue, staff are very friendly, food and drinks are good and it's open until 3.30 during the week and 5.30 Fridays and Saturdays - yes it does have music later in the evenings.... This place is just what the area needed... Try it...!