We moved from Battersea/Clapham to Streatham Hill for the same reason and we love it. Much more space, excellent green spaces, good transport links and well placed for good local schools as well as private schools in both Clapham and Dulwich.
The ABCD roads, Telford Avenue Conservation area and streets around Tooting Common are well regarded, close to the station and have good housing stock. Personally I’d stay closer to Streatham Hill than Brixton Hill in that area.
There are also some lovely houses and neighbourhoods closer to Streatham Common (where Streatham station would be your closest transport link). The streets off Valley Road are nice - the Streatham Wells area - as is the south side of the Common (Covington Way etc.)
Hope that helps!