Strange.. I added my infant when i booked on the website for united which includes a SK and two Lufthansa flights.

Infant ticketing is a nightmare though.. for some reasons airlines just can't figure it out.. it should be so easy, click a button add infant, it calculates fare and ticket issued...

I have used alaska miles to book business class flights for my wife and I and then added the infant later since alaska wont ticket infants on partners. It involves hours on the phone with Fiji air and Qantas. The fiji air segment was super smooth.. showed up at the airport. .she was on the ticket.. not problem get on the plane... the Qantas one .. not so much. Apparently even though i had a confirmation number and everything apparently it never ticketed... We stood at the check in counter for almost 3 hours while they tried to add it. The check in agent had to wait on hold with reservations just like you and me.. it was wild.. All while the line behind us got longer and longer since they only had 2 agents working .. and we took one for 3 hours... Apparently when i talked on the phone to get it ticketed months before the agent booked her on MEL-SYD connecting to MEL-LAX.. instead of SYD-MEL-LAX... and then it never ticketed.. We actually missed our SYD - MEL but thankfully had booked in a long layover in MEL to go to the First class lounge so we got the next flight and still made our connection. So i wish you luck...