Hi OP
i understand your concern about the lack of the A star in FM. His grades are excellent, for sure - no disputing that. But I think you are being realistic and, the fact is, every year, more are turned down with all A stars (even 4) than made offers. And yes, if you are applying from a super selective, the bar will be higher. This is the truth of it.
Might there be an opportunity for him to do another mini-exam or something in Sept for the FM to see if his prediction could be raised to an A star?
If not, it is what it is and it's still possible he might get into Cambridge, LSE or Warwick. He may perform really well in the Cambridge admissions test for Economics. Also the interviews.
As for the PS.... LSE put a massive amount of importance in this. Has he looked at the specific checklist for the PS on their website and tailored his PS, making sure it covers all those areas? This is really important for LSE. All they have to go on is the PS.
I think at Cambridge, the PS is less important in itself, because they will have the chance to meet him at interview and look at his performance in their admissions test. Also there is the SAQ. They may ask him to send some essays in too? So the PS is just part of the "jigsaw"in terms of their admissions. I'm sure you know this anyway. But - for Cambridge, please make sure his PS does not just list things he has read or done. Everything he mentions needs to be evaluated / analysed. He shouldn't waste space with anything non-academic.(sports etc).
So it doesn't matter that he didn't win an essay competition - what matters is specifically what he learned from the research and the prices of writing it. Did it inspire him to consider something else he sound not otherwise have been aware of?
Don't forget, Cambridge also have their own extra mini PS (the SAQ) where they can say why their course in particular interests the student. He needs to look at the actual modules on the course and tell make his wider reading / essays etc relevant to a couple of those.
Great that he's done an EPQ. If he got an A star in that, it's 30 extra UCAS points in value and Cambridge do like EPQs (not sure about Warwick and LSE but I'm sure it helps). Not everyone will have done an EPQ.
Bath is an excellent uni and offers Economics and will give offers a grade lower for an A star or A in EPQ. Worth bearing in mind.
Sorry go ramble, but I've just gone thorough it with a DC and the above are things I wish we'd known at the start of the process. Very best of luck!!