Based on bitter personal experience at airports the system may not allow you to enter France if you don't have the correct validity.

My wife (US citizen) was denied boarding for Iceland from UK on pretty much the same basis as you describe. The Passenger Information pre-flight requirement noted a short passport validity and would not permit the gate agent to issue a boarding card. Cue gnashing of teeth etc.

I endorse earlier comments - not worth the hassle. Get a new passport