1) API is a requirement on the airline, not the passenger.
2) API is not always related to immigration checks - for example not in all EU countries (yet).
3) AFAIK, API is not a requirement for flights involving Thailand, where the other countries involved also don't require it.
Of course, where one or more countries on your itinerary do require API, it will need to be collected before you can check in.
But, if you don't submit API yourself, the airline will just collect it when you present yourself for the flight, by swiping your passport. And it's much easier for a machine to read the MRZ of your passport(s) and fill in the data automatically, than to have to type in the information yourself - especially if you haven't memorized your passport information!
If you don't particularly care about checking in as soon as possible (T-24, T-48, or even at time of booking depending on airline) then you might as well save yourself the bother. If you need to check-in ASAP because you want to choose a seat, or because your airline has a history of overbooking and bumping the pax who were latest to check in, then obviously entering a few fields of data is not particularly difficult.
On the itinerary mentioned in the OP, it really doesn't matter which passport is used! If the airline is unhappy, they will just ask to see the other passport, and if the API needs to be altered it will be done by the check-in / gate agent.
I occasionally transit Thailand, on separate tickets, when flying between my countries of citizenship (A and B). On the flights to and from A, I check-in with passport A. On the flights to and from B, I use passport B.
I only enter Thailand with passport A. Nothing ever comes up at Thai immigration when going to/from B. Sometimes at BKK check-in, I present passport B for my flight to B, and they will ask why there is no Thai entry record. I then show passport A, and that's the end of the matter.
Sometimes, the API for my flight to B contains passport A's info because it is a through ticket from A. Country B doesn't seem to care about this, provided that I enter B with its passport.