I live in central London and am learning to fly from Biggin and doing the journey almost every day, if I drive, I find the A2212 the best route as few traffic lights and generally lighter traffic, it looks counter intuitive on the map as it is further east, but it avoide catford and just skirts the edge of Bromley. The only sticking point is Grove Park to be honest.
Otherwise, I agree with the train, Victoria to Bromley South and either the 320 or the bus.
Either way driving in South London has always been a royal pain the rear end!