That is specualtion at best; how can
you possibly know what is "far easier and better" when you don't know where the OP's 'person' is located, when travel is, or the status of the current passport application. CBP clearly anticipates this situation and can handle it:
https://help.cbp.gov/app/answers/det...e,-citizenship
How can you possibly know that it is "far easier and better" to work this out with the Department of State (overseas consulate) than with the Department of Homeland Security (CBP at the airport).
(Oops-- phrasing the question by cabinet department almost answers the question because DHS is . . . well, DHS). But in all seriousness, there [at least] two ways to go about this, and I don't claim to know which is better without knowing the situation; why do you?
By the way, the carrier isn't going to say anything unless there isn't an ESTA on the VWP passport-- and it has been established that you can get an ESTA as a USN-- so your #3 goes out the window.
OP can make his choices as he wishes. To me, time is money and and the less I have to deal with paperwork uncertainties, the better off I am. In the final analysis, his relative will wind up being admitted.