The council is in favour. Andrew Gilligan - the outgoing mayor's cycling commissioner - has also made positive noises about extending the scheme to cover more of Southwark.
I suspect the sticking point is how much money TfL wants the council to contribute to the overall cost of the extension.
Several recent planning permissions for major developments in Bermondsey east of TBR have included financial contributions to install docking stations, so some of the background work is quietly being done.
It will happen - it's more a question of when than if.
It'll be something for the new mayor and whoever they appoint to lead on transport/cycling matters to get their teeth into.
It mirrors the situation with electric vehicle charging points. There's a conspicuous lack of EV charging points in Southwark when you look at the map at https://www.sourcelondon.net/
This is despite a lot of new developments being built, and until 2015 grants available to local authorities, to install charging points. I had protracted correspondence with Southwark in 2014-15 but it got absolutely nowhere and I decided to give up my EV as a result.
With businesses and residents increasingly using EVs and air quality causing ill health to so many, 5+ years of almost no action on EV charging points and bike hire points (yet the quietways dreamt up and delivered in the same timescale) makes no sense. I think the highways department could learn a lot from the parks department about how to deliver public realm improvements!