Your son is clearly a bright boy and his current school doesn’t want to lose him.
You don’t say whether he is at a state school now, or at an independent school without a history of sending boys to Eton, which can’t or won’t advise you very well. If it’s the latter they also won’t want to lose his fees…
It’s firstly worth saying, as his current school doesn’t appear to have done so, that not all
scholarship processes are the same. KS and Election are insanely difficult, close to A level standard, and really do require specialist teaching. But other public schools have academic scholarships that are a lot more attainable. Some will stick closer to the CE curriculum, some might involve an exam plus interview, and for most the competition will be a lot less serious (as convention dictates you can’t enter for more than one scholarship).
To be brutally honest, it is not too late to change schools and I think you should at least consider it. Most prep schools don’t start scholarship teaching in earnest until the start of Y8 so you have time. You could still get four terms in without incurring withdrawal fees if he’s at an independent school already, or you could even move him this Christmas if he’s at a state school.
As a first step you should go and see some headteachers of preps who have sent at least 2-3 boys to College (KS winners) in the last three years Explain the situation, show them his school reports/exam results, and ask them where they think your son would have a decent shot. They will be frank, because it doesn’t serve them to take a boy who can’t cope with scholarship work, and will give you some useful advice as to your options.
In terms of your specific question about papers. Clearly there’s no point taking up Latin at this stage if he hasn’t done it already, just for the sake of doing an extra paper. You only do additional optional papers if you are already strong in multiple subjects and genuinely aren’t sure which three will turn out better on the day.
Re tutors. I think you’d need someone who has previously tutored someone who has won a KS, and who therefore knows what good enough in this context looks like. Your average A level tutor would not have a clue. They would also probably be puzzled by the lack of any formal syllabus or set texts.