OP, what do you mean by Battersea end? Battersea square? Power station? Queenstown road? Are you looking to move or to stay put for secondary?
Unlike other councils, Wandsworth published the admission results of each school, including how much the waiting lists moved from March to September, so you'll find plenty of information online.
St John Bosco and Harris Battersea are both undersubscribed. Many people think that's because it attracts many families from very rough council estates nearby, and the environment is rough.
Chestnut Grove in Balham is considered a very good school, but you're unlikely to get in based on distance; you might have a chance based on the art or language specialist tests.
Graveney in Tooting: your child needs to do well in the Wandsworth test - no chance by distance.
Ashcroft Technology in East Putney is likely too far, but they have a specialist technology test. I'd never send my child there. The ethos is very similar to Holland Park, Mossbourne etc. There is strict but fair, and there is batshit crazy; a headteacher who gives you a detention if you cycle to school falls in the latter category. I'm not making it up, look it up online, you'll see it on their policies.
Hurlingham in Fulham doesn't have a 6th form. Would you have got in by distance?
Harris Wimbledon has a specialist test for music.
Have you looked into St Cecilia in Southfields? Some religious places, and some places based on a music test (harder and more thorough than the Harris Wimbledon one)
Bolingbroke is almost impossible to get in unless you come from a feeder school. I think no one has been admitted from a non-feeder school for 3-4 years, but you can look up the details on line.
If you are close to Lambeth, bear in mind that Lambeth has many, many undersubscribed secondary schools. Harris Clapham is probably the closest to you.
I cannot tell you much about religious schools, other than mentioning St Cecilia.
It is an utter disgrace that in this country ca 1/3 of state schools are religious, and that everyone's tax money is used to prioritise those who say they are religious in the provision of such a crucial service as education. Shame on those who support this travesty of public service!
Imagine if healthcare worked like this; imagine if 1/3 of hospitals gave priority to those who are religious, despite being funded by everyone's tax money! We wouldn't accept that bs with healthcare, why do we accept it with education??