Hi there, we also live in Putney and had to make a similar decision several years ago when our eldest son was little. We went to look round Merlin, Hurlingham, Colet Court, Fulham Prep, Thomas's Fulham, Thomas's Battersea and tried to gauge whether we wanted pre-prep then separate prep or a school that went through to 13+. We chose continuity to 13+ in the end in the school that just 'felt right' for DS1 as we walked through the door - he was 3 by then. And we had to find a way round a full registration and waiting list once we decided...!
So you may find that as you look round nearer the time, you just get a feel about where your son would be happy which is what ended up governing our decision rather than entry/exit points. As you've registered him at lots, you'll get lots of choice!
Also, when they're little, you really can't tell where they'll end up later on - DS1 is off to Eton next year and until Yr5/6, boarding was simply not on the cards in anyone's mind and certainly not the school I was probably most prejudiced about....
11+ for boys is very limited if you want the big day names round here - they take very few compared with the numbers of boys who live in the area. So to me, a school ending at 11 is risky unless you love it.
Kings takes approx 56 boys at 7+ and 10 at 8+, Colet Court takes about 36 at both ages. Westminster Under, I have no idea about but is ferociously competitive at 11+. If you really want the big name day schools, you'd be best with Merlin then heading on at 7/8+ or at a school ending at 13+. You could ask both Prospect Hse and Hurlingham's registrar for a breakdown of boys' exit schools at 11+ - just check that when they say 3 into Kings, 3 into Colet and 3 in Westminster, it's not the same 3 boys.
It gets easier to choose as they get nearer to Reception age and you get a better sense of their nature. Hopefully you won't have to pay a deposit till nearer the time or at least it will be refundable with a term's notice?? Good luck!