Not initially. You do need an AHC to get into the EU (although Ireland is different, but that's a long story).
Once you are in the EU you can go to a vet and get an EU Pet Passport. This is because the passport is only a record of rabies vaccination, it has nothing to do with nationality.
France is more difficult because the dog ”should” be registered on their chip database first and that requires a French address and a period of residency. But, as said earlier, a quick trip to Belgium gets round that issue.
Two important things.
The vet will enter the date they read your dog's microchip into the passport. The date of the rabies vaccine mustn't be before that date. Rabies is a dead vaccine so having a booster early shouldn't cause any problems. That takes three weeks to be fully effective, so if you are travelling back before 21 days, you would just use the AHC to come back.
The other thing is that only vets in the EU can make further entries in the passport. So future rabies boosters need to be done in the EU.