With a family of 4, presumably you'll be after a 3+ bed house? 2k pcm isn't very much by London standards but there are some lovely suburban areas commutable to Charring Cross that will work with your budget. Boroughs of Bexley and Bromley will offer direct trains to ChC from zone 5. I'm not sure about fee paying schools but both, particularly Bexley is amongst the best local authorities in England for state schools. It's also got some of the lowest crime rates in Greater London - basically a great family-orientated suburb.
I live in the borough of Sutton which is also in zones 5 and 6 of South London and might be worth a consideration. There are no direct trains into Ch C so the commute would be less good (still very reasonable, though). Again, very family-orientated and super-safe with state secondary schools at number 1 in England for attainment. Excellent for city and outdoorsy lifestyle (countryside in the south of the borough, hills a 20-min drive and the beach under 1h).
I'm not much of a commmuter town expert but would say to factor in commuting costs. An annual ticket from zone 5 is 2.3k whilst commuter town commutes often set you back 5-7k per year.
Also, the reason I mention state schools is that good ones are abundant in these budget suburbs. There are fewer of the independent sort as they tend to be more common in more expensive areas so on 2k pcm you'd need to squeeze into a 2-bed flat.
The only area I can think of that would be commutable for DH, offer a selection of fee-paying schools is the south of the borough of Croydon (South Croydon, Purley etc) but I'd find it too close to Croydon proper for my liking.
Alternatively, hopefully your relocation person will have some bright ideas about commuter towns if you feel strongly about fee-paying schools.
On the sports front, gymnastics and football options should be easy enough to find most places but basketball is going to be a bit more of a challenge as it's very niche in England.