London Bridge has always been a busy station as are both lines but by 9.00 it is starting to calm down a lot. In fact most of the problems with the main line (train) station are happening in the evening rather than the morning.

You shouldn't have any any problems at all. However it is always good to have alternative routes.

An alternative might be to walk over London Bridge to Monument or Tower Bridge for Tower Hill for the District line where you can connect to the Piccadilly at Earls Court, Barron's Court and Hammersmith (the latter two just involving crossing the platform)