The problem with putting your US passport API details in a flight going from the US to any country that requires some kind of electronic travel authorisation or visa (and you also hold a passport that exempts you from that) is that the airline can block you from checking in as for them it looks like you are not eligible to travel. So if that's the case, in the API you always need to put the passport details you want to use to enter the country you are travelling to.

I had an interesting case a few years ago when I was travelling from the US to Germany via Canada. I did put in my NL passport details, and the system blocked me from checking in. I did not realise that Canada changed their entry/transfer rules for EU passports and now required an ETA. Had to call in and give them my US passport details - which would cause another problem when the EU finally implements their ETA scheme...