To be honest your budget is good for lots of the areas mentioned on here. Slightly less so for a few of the locations but still possible.
The areas mentioned are so wide and vastly different that realistically you need to go to all of the areas and decide your top location and where you actually want to live otherwise you’ll drive yourself mad!
Factor in future proofing with children/family life as the stamp duty is obviously weighty at this price point.
Then drill down from there suitable houses within your preferred location(s).
For instance I know really well the locations mentioned of
Wandsworth, Dulwich, Sydenham, Forest Hill, HO, bits of Ladywell, Crystal Palace, Penge, Kent House, New Beckenham and actual main Beckenham.
My choice out of all the above would be Wandsworth, Dulwich and Beckenham. But for your budget you would get different houses, square footage and gardens in each. They are all expensive but Beckenham is very different to both Dulwich and Wandsworth.
Then when you do the same Blackheath, Greenwich, Peckham etc exactly the same happens so you really need to like the vibe and amenities and style of roads, housing to be able to decide. Factor in transport links etc etc. otherwise your search area currently is just too vast.