Update:
Took over 50 minutes total to book two round trips to Europe for my wife, infant daughter, and I.
About 30 minutes through she gave me the total taxes which I realized were incorrect. I asked for an explanation. She forgot to book my wife a seat. She said, "It's just you, correct, and your baby?"
WHAT? "No, I just gave you my wife's name and birthdate not six minutes ago!"

Long story short, she booked my wife, but was unable to book her on the same reservation. So now we have two separate reservations and I have no idea who has the lap child.
How, how, how does this company not run into the ground?