Congratulations, it appears that your child has dual US and French citizenship. This will gain him various advantages later in life, and you will do him a favor by ensuring that he has full access to both countries' protections.
If your schedule permits, you should obtain your child's French passport before you go. If you don't have time, you still should obtain it. This can be done at the French embassy or consulate serving where you live in the US. At any rate you must do this before he turns 18.
Even if you don't have time to get his French passport, your child can still travel on his US passport. As you may know, US citizens can enter the Schengen area visa-free for 90 days out of every 180 day period.
EU citizens visiting other countries in the Schengen area technically have a similar requirement, but EU citizens also have treaty rights which are activated in certain circumstances (e.g. moving for the purposes of work).
Of course you'll likely have the extended questioning for which you plan to bring the birth certificate. This issue goes away if you are able to obtain his French passport before your trip.