As an overabundance of caution, I would take with a great pinch of salt what the local consulate told you, unless you have it in writing. At the very minimum, I'd apply for the U.S. passport for the 1 y.o. before your trip and get the receipt for it. That's one more piece of evidence you can show the I.C.E./C.B.P. agent. It'd only take one in a bad mood and/or stickler for the rules to derail the whole trip. No guarantees that your trip may not be ruined/derailed anyway with a passport application but at least its one more piece of evidence you intended the 1 y.o. to travel with a U.S. passport but couldn't get one in time.

The other thing, which I do not recommend, is to not mention anything about the registration of birth but a stickler may well document and record that you did not intend for your child to have U.S. citizenship.