Since you are planning to move house, you might as well make sure you have what looks like one good sure-fire possibility and move within distance of a school which is currently good or on the up.
What sort of area were you thinking of? Some of the Dunraven catchment is in relatively less-expensive areas of Lambeth, as is some of the area for Elmgreen. (Tulse Hill / W Norwood) In Dunraven Catchment
Norwood School is improving fast, and Norbury Manor Girls would be worth a look. Norbury is much cheaper than Streatham.
You can visit schools Open Days before the year you need to apply, so you could go to the ones this summer.
Look at the schools, study their admissions criteria, and think about the area for all the areas you are interested in / could afford. You don't want to go to all the trouble and expense of moving house to find that you have moved into a schools black hole, (Streatham Vale for example) and are allocated an under-subscribed school a longish distance from where you live.
Your LA does have to find you a place, but the reality is if you are just too far away from a school you really like, you may be allocated a school which has places - i.e is unpopular.
If your current local secondary is good, is the suggestion to stay put until after transfer a possibility? Chestnut Grove on the Lambeth / Wandsworth border has a great reputation and is 'Outstanding'.
S London schools where you can get all or some places not dependent on distance include Kingsdale (lottery and scholarships in music and sport) Norwood (music), Harris Crystal Palace (not sure of the specialist places, complicated, some lottery places).