Dont correct the passport info until after trip is over. Make it simple. For now Timmys birthday according to his passport is jan first. And he travels as if his bday was jan first.

When he finishes the trip and safely back in the usa ammend the passport.

I cant imagine ie american airlines ever coming back and saying 'wait a minute last time you bought a ticket for timmy for his paris trip last year his birthdate was jan first and now it jan second.'

Even if timmys erroneous bday was part of his frequent flyer account a simple fax of the corrected passport of his bday info will fix that in a jiffy. People mess up simple info like that all the time. But do it after the trip is over.

Call the airline.
Match the current birthdate of record with tsa info needed.
Fly.
Then when you have time officially ammend the record.
No fuss no muss. Airline doesn't care. Government doesnt care.

All they care about is if the secure flight reported in this specific reservation matches the passport used for travel. It has no bearing to previous nor future travel. Only the flights in question.

I can only imagine headache and government red tape trying to fix this so soon of your trip. You will prob spend tons of money and no guarantees that the state department will come thru in time or gasp maybe get it wrong again.

Thats what i would do but to each their own.