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This list of English endowed schools by county from the early 19th century is organized by county/shire, including Surrey (location of Brixton) and the adjacent Buckinghamshire and Middlesex/London counties, which include among others the town of Eton and its eponymous College.
Surrey:
-- Wilson's Grammar School, Camberwwell (Most likely possibility - see below)
-- Farnham Grammar School
-- Royal Grammar School, Guildford
-- Kingston Grammar School
-- Saint Olave's Grammar School
-- Saint Mary Overey Grammar School
Berkshire:
-- Abingdon Grammar School
-- Childrey Grammar School
-- Newbury Grammar School
-- Reading Grammar School
-- Wallingford Grammar School
-- Wantage Grammar School
Buckinghamshire:
-- Amersham Grammar School
-- Aylesbury Grammar School
-- Buckingham Grammar School
-- Eton College
-- Royal Grammar School
-- Marlow Grammar School
Middlesex (Now wholly incorporated into London County I believe):
-- Enfield Grammar School
-- Hampton Grammar School
-- Harrow grammar School
-- Highgate Grammar school
London:
-- All Hallows Barking Grammar School
-- Charterhouse School
-- Christ's Hospital School
-- Mercers' Chapel Grammar School
-- Merchant Taylors' School
-- St. Paul's School, London
-- Westminster School
Wilson's Grammar School was located in Camberwell from founding until 1975. With the centres of Camberwell and Briston barely 2 km apart, that would likely be the school closest to Brixton at the time of interest.
Counties of South-East England:
Many thanks to the commenters below who assisted in correcting my faulty knowledge of South England geography.