If there was a 381a that did the normal route (going the big loop round rotherhithe) and a 381b that turned west down southwark park road/grange road, right into TBR and left into Tooley - that would work for me! It would not be a 'new bus route' but a bifurcation of an existing one - that word alone would excite Boris's juices and get him to throw his towel into the fight for us.
If you are not elderly, or handicapped, then walking is far the best way to get from Bermondsey to Borough because I am 50 now and lifelong asthmatic and have other health problems but after being attacked on the tube in the middle of the day with a whole bunch of people watching who sucked in their breath as they watched me being repeatedly punched but did absolutely nothing I now walk everywhere from Bermondsey to Waterloo or one mile or more around and only take trains if I have to go further afield and there is no parking and my health is much better as a result of all the walking.
I can now walk from Abbey Street to Borough in ten minutes so sitting on an expensive, slow and unpleasant bus if you aren't vulnerable seems strange to me with all my problems.
Hurrah for the lack of security on the underground in spite of all the cameras that let you get assaulted and your attacker walk away which we all pay so much for.
At least you you get more exercise by avoiding the Tube.
Oh, and if you aren't elderly or unfit then walking is far better than the busses too.
You can spend as long waiting at the stop as you you can walking to your destination over short distances and it is free and safer.