Streatham Hill / Tulse Hill would be a very good bet.
Dunraven as mentioned, and nearby Elmgreen are very good schools, and if you could get a house on the Tulse Hill side of Leighham Court Rd and close to Dunraven, you would be within good reach of both. Nearby there is also the City Heights Academy, which I hear has excellent teachers. It is new, so hasn't developed its reputation yet, but it is a London school In a borough where all schools, on average, outperform the national average (I think).
You would also be in a good triangle of Primaries. Streatham Wells is a fabulous little school, Hitherfield is doing very well, very popular, and Dunraven also now has a primary. It is new but should be accepting 7 year olds now or next year. A bit further along is Sunnyhill Primary, also a perfectly good school. Hitherfield and Sunnyhill are quite big, so places come up quite often. Also within walking distance of the house linked would be Julian's Primary - very good.
Streatham Wells has only one form entry so maybe not so many opportunities of a vacancy, but this area of S London has high mobility , so there is a turnover of places.
There is good transport from Tulse Hill Station and Streatham Hill station, and good buses.
Compared to other similar areas of London the prices are not as high (but obviously not cheap!).
Does your eldest have any exams or courses that match the curriculum and exams here? Has she actually been studying GCSEs? It may be that she could go into Yr 12. I think they can start A level courses a year later.