That only applies to the Southern (Central) side. Diesel and, with regards to charters, steam services can run into the Southeastern (Eastern) side. That said, it all depends on how many services Southeastern need to run, and whether HSTs and other GWR stock can be cleared for the route (probably more about short swing link / long swing link issues than other dimensions; believe it or not, the lines via Grosvenor Rail Bridge into London Victoria were originally designed to accept GWR broad gauge freight anyway).
Except the bridge got rebuilt in the 1950s.
GatEx 442s have run via Herne Hill on diversion during the Balham closure a few years ago so should be able to do the short Battersea - VIC stretch.
However SET will have diverted services into VIC due to London Bridge works...