Luckyone1, I’m intrigued what hurricane wind had blown you up this Eton way out of the blue at this extremely late hour? I mean, once upon a time, not very long ago, parents were already preparing for Eton even before their son was born!
I know this is not of much help to your cause (sorry) but barring a miracle, it is almost impossible to win a King’s Scholarship at this very late stage with absolutely no prior preparations at all. I believe whoever that advised your son to take the KS exams probably meant well because that is the only way of gaining entry at age 13 seeing you have not registered your son before age 10½ and assuming your son is not eligible for the New Foundation Scholarship (for state school boys).
Advising to sit the KS exams is one thing but realistically succeeding in winning one is quite another thing. Please be under no illusions - the King’s Scholarship exams according to an Oxford don are the toughest for a 13-year-old s(he) has ever seen. The level is of an Oxbridge First Year standard. Quite a few Oxbridge graduates have said they couldn’t do some of the questions.
Potential KS winners would have been preparing for the exams for at least two years in a proven prep school. These same prep schools have more or less become feeder schools to College over the years. They are usually situated within a 60-mile radius (or less) within London. At the top of my head - Westminster Under School, Summer Fields, Papplewick, Dragon, Sussex House, etc.
In the last eight to ten years, increasing number of successful boys come from the Far East including Singapore, Thailand, Korea, Japan and China. Unless the rule changes, I foresee College being occupied by at least 50% ~ 75% in the next 25 years by boys from the Asian continent.
Coming back to your question, OP, I think it is totally pointless to move your son to a prep school for 1½ term. For a start, a proven prep school like those mentioned above would not be doing much teaching in the final term. This is a time for ironing out the creases that may still be lingering in a boy’s mind and to refine all that has been taught and learnt.
As for personal tutor, again, I don’t think this is of much good. Sorry! A personal tutor may be a specialist in his/her subject but it is very unlikely that s(he) would be up to scratch with the other subjects at KS level. I have often heard of this phenomenon of personal tutors specialising in KS exams but in real life, I’ve never met a single KS who has had a personal tutor. Let’s just remind ourselves what the examiners are looking for:
”the examiners are keen to reward boys who show real ability, even if in only a limited field.”
In short, a boy who needs a personal tutor for whichever subject sadly, has no “real ability”. Worse still, his life would be a misery if by a twist of fate he did find himself landed in College living in the midst of ferociously high-flying boys . But if you still want to try personal tutors, I wish you luck and hope a miracle does happen.