As far as getting back into France is concerned I am not sure how much checking is done since you're inside Schengen. However check the small print, particularly regarding dates on your vaccination document, and read the small print on the french govt web site.
Madame and I are in a bit of trouble over this when we go to UK in august. I am over 65 so eligible for a booster, she is not. Her vaccination status runs out 09/08, messed up by both of us having the dreaded lurgy in april. We are set to recross 15/08! It looks likely that without PCR tests done in UK she won't be able to get home! (She is french, I'm british but that has absolutely nothing to do with it, the rules are simply french regs). If your vaccination status is not complete when you come back into France then expect a demand for tests, either lateral flow or PCR, same for everyone.
Oh and ..... don't catch the lurgy,even mild, it messes the dates up!