Lost in the fog of warm and fuzzy discussion of family values is the following overlooked point that the OP made:
Definition of K-1 visa acccording to US Dept. Of State:
Nonimmigrant Visa for a Fiance(e) (K-1):
If you are an American citizen and you want your foreign fiancé(e) to travel to the United States to marry you and live in the U.S., you must file Petition for Alien Fiancé(e) in the United States
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Given this constraint added to the non-deterministic date of the rescheduled interview, it is pretty much an either/or scenario.
Definition of K-1 visa acccording to US Dept. Of State:
Nonimmigrant Visa for a Fiance(e) (K-1):
If you are an American citizen and you want your foreign fiancé(e) to travel to the United States to marry you and live in the U.S., you must file Petition for Alien Fiancé(e) in the United States
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Given this constraint added to the non-deterministic date of the rescheduled interview, it is pretty much an either/or scenario.