We should not forget about the Honor Oak Park area, which is close to Stillness and Dalmain - both very good schools. As are others in the area, including my old school Eliot Bank, which is on the Sydenham border.
Basically, there is no shortage of 'good' and 'excellent' state primary schools in the local area.

If you are looking for a 2 bed garden flat then I think looking in the Honor Oak Park or Perry Vale area (west of the railway) are just as good, although the museum and future swimming pool is a little further away.