They might, they might not; it is not a forgone conclusion.
What makes you think this would cause a lengthy go-around with CBP? Do you think they've never seen an expired document before? I'm sure it would take a little longer than a standard entry, but a "lengthy go-around" sounds more like speculation than reality.
I personally know of two recent instances of travelers using expired travel documents at a US POE with minimal delay. By minimal, I mean that a supervisor was called and said that the expired passport was acceptable, and it was a normal entry after that. Both were US Citizens. Using an expired US passport to enter the US just isn't that big deal, particularly close to the expiration date.
So, based on two instances of however million/year, you're prepared to advise OP to travel on his French passport and on his expired US passport back to the US (and hope that the carrier accepts the expired doc. at departure)?