Is there a USA Passport Agency near you? I think that's the term for the wonderful federal offices in several dozen USA cities that can issue same-day passports. (I don't mean post offices that handle passport applications.) If you can travel to one of these, make an appointment and ask them what documents to bring? (Is an official birth certificate enough? Do you need both parents and/or the child to be present for this?) What payment forms do they accept? Proof of immediate travel for same day or next day service probably requires a hard copy of the itinerary (maybe including hotel reservations too?) and plane ticket receipt with ticket number.

Personally, I would feel more confident about fixing the error this way, in person, than by mail with or without the "help" of a passport and visa service that might or might not be knowledgeable and competent with issues like this one. To avoid worry--and since the passport will need to be corrected at some point--I would try to do this ASAP. You have time at this point, so why take the risk?