I agree; the only place where you have significant numbers of people of that sort of class going to day school is in London (Westminster, St Paul's).
Unless - just perhaps - Lewis was Delicate (like the twins) for a period as a boy (post rheumatic fever - a splendidly 1930s literary illness, perhaps) and spent a term or two at the local Very Old Foundation (ie 1500s) grammar school being a day boy before being packed off to Marlborough or wherever? Admittedly, private tutors might be more likely in such circs, but Sir Charles might have thought them namby-pamby or otherwise suspicious.
It's a stretch, but it's my settled belief that practically any backstory can be prepared to work with sufficient thought about how it would affect the characters (in this case, Foley being bullied as the outsider, Anquetil coming to his aid as a fellow outsider albeit for a different reason, Foley choosing a different way of being an outsider, namely the outrageous pranks route rather than the protective colouration route...)
Or Sir Charles may have briefly saddled Lewis with a Manic Pixie Dream Girl stepmother, who decided that the bourgeoisie were the only people with true Values (like the irritating artist in The Documents in the Case, who then spends most of the rest of the novel complaining that his bourgeoise lover won't live with him in sin, because that would be Wrong and People Would Talk) and he got sent to the grammar school for a term or two before Sir Charles got irritated with Lewis's vowels and his wife's politics, both, and ended the experiment.
Re: Friendships
Unless - just perhaps - Lewis was Delicate (like the twins) for a period as a boy (post rheumatic fever - a splendidly 1930s literary illness, perhaps) and spent a term or two at the local Very Old Foundation (ie 1500s) grammar school being a day boy before being packed off to Marlborough or wherever? Admittedly, private tutors might be more likely in such circs, but Sir Charles might have thought them namby-pamby or otherwise suspicious.
It's a stretch, but it's my settled belief that practically any backstory can be prepared to work with sufficient thought about how it would affect the characters (in this case, Foley being bullied as the outsider, Anquetil coming to his aid as a fellow outsider albeit for a different reason, Foley choosing a different way of being an outsider, namely the outrageous pranks route rather than the protective colouration route...)
Or Sir Charles may have briefly saddled Lewis with a Manic Pixie Dream Girl stepmother, who decided that the bourgeoisie were the only people with true Values (like the irritating artist in The Documents in the Case, who then spends most of the rest of the novel complaining that his bourgeoise lover won't live with him in sin, because that would be Wrong and People Would Talk) and he got sent to the grammar school for a term or two before Sir Charles got irritated with Lewis's vowels and his wife's politics, both, and ended the experiment.