After a lengthy back and forth between TflL and Lewisham Council as they argued about who owned it (Lewisham Council, it turns out), I have now been advised that:
Understandable, of course, but rather disappointing.Unfortunately the [Council's] Structural Engineer has confirmed that whilst we apparently do indeed own the sign, we do not currently have the funds to maintain it. I am advised that the repair in this instance would cost around £600. As I am sure you are aware highway maintenance budgets, in common with other Council budgets, are under extreme pressure at present and it is difficult to justify this expenditure as it would simply mean that other more urgent issues could not be addressed.
The full saga is available at https://www.fixmystreet.com/report/727677#update_832328