For changing between National Rail and London Underground at London Bridge you will need a Zone 1 - 4 travelcard.
Wih your current travelcard you would need to use the London Overground service to Canada Water.
Unfortunately the zone charging is about what zones you travel THROUGH, so you need a card for Zones 1-4. It won't be accepted at London Bridge.
You will need to go through 2 sets of barriers at London Bridge when you change trains and your travelcard will not be accepted because London Bridge is in Zone 1. You will need a Zone 1 to 4 travelcard.
www.tfl.gov.uk
www.nationalrail.co.uk
If you are travelling through Zone 1, you will need a travel card that covers that zone.