Well I have a Z1-3 travelcard so it's irrelevant to me, but the question came up for Mrs. Stifle (who would be using Oyster PAYG). The problem is that there appears to be long stretches of the day during which there are no trains on the Blackheath/Charlton stretch or v/v. The NR Enquiries site says you need two tickets to go via London Bridge, which is probably a hint.
Seems rather silly to resort to the Routeing guide, but Westcombe Park has routeing points Charlton and London group, and Kidbrooke has routeing points Charlton, Dartford, Lewisham, and Slade Green. As they have a common routeing point, that means that "the permitted route is direct via the shortest distance from the origin to the destination over which a regular scheduled passenger train service operates", which would indicate that you must go via Charlton (unless you get one of the Southeastern circle services from London to London via Greenwich, Slade Green, and Barnehurst, if that combination even exists).
Didn't think of the DLR, although that would presumably increase the fare from a National Rail fare to a through fare?
Final thought: I seem to have heard of a three-mile rule whereby you're allowed to take a route up to three miles longer than the shortest route. If that's valid, would that allow travel via London Bridge? I don't have any distance tables handy.