Correct procedure would have been to apply for a U.S. passport at the U.S. embassy. IMO the embassy would have given the applicant the run around. I am not surprised the N-600 route was tried.
Once N-600 is denied, one cannot apply for it again so I would do everything humanly possible to make that appointment.
* Definitely try an emergency passport appoint with embassy.
* Canada, Mexico, and U.S. require visas for citizens of Antigua and Barbuda. I would apply for tourist visas at all 3.
If U.S. does not approve a tourist visa in time, maybe Canada or Mexico will. If so, show up at land border with all the USCIS N-600 notices in hand, and after long secondary be admitted to the U.S.
* While waiting for a visa or passport appointment, since no visa needed for Bahamas, and I believe Turks and Caicos, Aruba, and Bermuda, and so each of those has U.S. CBP pre-clearance ports of entry, I would fly to one or more of each, and try to board a flight to U.S., again showing all USCIS notices to CBP at those countries. The challenge is getting the airline to issue a boarding pass.
* The closest living relatives in the U.S. needs to get their U.S. senators to help.
If you cannot make the N-600 appointment reschedule.