I am in literally the same boat. Booked for me from my account, wife booked from her account and now we have problem booking for our child. Talked to at least 5 different people from the TK call center and no one was able to book it for me. The reason was always different from the previous. So far no one mentioned the rule I need first to finish my first award flight. Also, it's not in their FAQ: "You can also arrange awards tickets for another person using Miles in your account.".

I filed a complaint through the feedback section and got a response the only way is to visit the sales office. And there's no sales office in my city...