If your son is a French citizen, he can enter and remain in France, period. All he needs is proof of it.
Since he is a French national by virtue of you being French, all it takes is:
When entering the UK and France, he should present these documents to facilitate passport control (if going to the UK by air, you should join the UK/EU queue at the British as well as the French airport).
That said, unless you actually want to visit the UK, don't bother channel-hopping. Like I said, your son can remain in the Schengen Area - and if he couldn't, then channel-hopping would not help, as "pure" US citizens can only stay for 90 days in a 180-day period.
As soon as he gets his certificate, he can get an ID card or passport
(TIP: a French ID card, which is free, is good as a standalone document for travel - including immigration control - in virtually all of Europe, Turkey, Georgia, parts of North Africa and much of the Eastern Caribbean. Combined with a US passport, it's unlikely to be worth the €86 for your son to get a French passport as well)