Luckily I was talking to a very helpful airline employee, and (I guess?) because it was the airline trying to book the tickets (with themselves and other airlines for other legs of the trip), I wasn't charged for each booked-but-ultimately-fictitious flight.
But even the airline employee seemed mystified. What would happen is she'd find and reserve the flights in her system, read me the whole itinerary, then get to the last step and say, "oh that's weird, <other airline> isn't accepting adding you to this flight." This happened three times, with different flights / different partner airlines.
In the end the only thing which worked was to book an identical itinerary, same day of the week and times, just one week early - implying that the only "real" flight out only runs once a week.