France New Covid rules - HELP!

There is no significant evidence of harm from having the jab 4 weeks post covid.
They changed the advice to 12 weeks partly because there seems to be limited gain from a jab soon after covid. You have to weigh this up against the impact of the potential hassle not being jabbed will have on your holiday.
If there is no PCR (and thus no recovery certificate) it is unlikely any vaccinatination centre will decline to give him his second shot.

The need for a PCR test was dropped when Omicron was at it's peak because of the sheer numbers and when there is a lot of Covid around, proportionally the numbers of false positives are relatively small. The flip side is that for those who are travelling a PCR can remain positive for a long time after recovery and that might be a problem if a PCR is demanded for travel so cynically I would suggest that if you have symptoms or a positive lateral flow get a PCR as that will both confirm the infection and automatically generate a recovery certificate at 10 days.

I have heard it said that if you report a positive lateral flow the recovery certificcate may take up to 29 days to show although I don't know for sure. My PCR and my wifes PCR showed recovery certificates at 10 days.