We brought a cares into NW angled lie flat seats when DD was 18 months old. The problem that we had was that the chest buckle ended up right at her neck (rather than chest) and it is not adjustable. We couldn't pull it down any further on the seat due to mechanisms behind the seat. We put pillows under her to bring up her height, but thinking back on it I am not sure that was a good idea- in an emergency situation the could fly out from under her.
Many of the completely flat business class seats have airbags in the seat belts making car seat use impossible. Not sure about UA or not.
We also had the problem on the angled lie flat seats of her sliding down while sleeping and the seat belt getting wrapped around her neck. The FA woke me with a great deal of concern one time.
What we ended up doing was bring ALL of the following and just figured out what worked: car seat (ended up gate checking it), CARES and Baby B'air toddler version. The Baby B'air worked great on the return when she was sick and only wanted to sleep with Mamah. I just strapped that on and let sleep on top of me. We only needed the car seat in Singapore for the taxi ride to the hotel (though we were glad we had it- it was a long and very fast ride).
You should have a nice trip. We really enjoyed SIN with our 18 month old and are returning this fall again.