I flew TATL with my eldest when he was 5 mo old. He was colicky, and was getting his first tooth on the plane. Despite all that, he slept through the flight and it was possibly the most sleep we had gotten in a while.
We had him in his car seat and I think that made a huge difference as we didn't have to disturb him for meals / going to the bathroom / etc. At under 3 months, most babies interests are eating and sleeping so you should have a fairly easy time of it. Personally though, I'd book direct and book a seat for the baby. You're going to be so tired at that point, the extra travel time can be really wearing on you. Also if you have a seat, you can drape a muslin blanket over the baby carrier to protect them from light / noise / maybe even some germs in the cabin and help them sleep better.
I'd also travel after at least the 2 mo vaccinations, when you're also past the point where you're supposed to take the baby to the hospital for any fever, even a low one.