Ah... I just realized something. I use the car seat rear facing because... welll... that's what I'm used to. (In California, the law requires rear facing until age 2 or unless your baby is way off the charts big.)
So the reason the seat in front could not recline on Southwest was definitely because rear facing. As for what the seat didn't quite fit maybe it was also because rear facing - it's not unusually wide or anything. In fact, it's marketed as being great for travel (especially due to weight.)
Hm... maybe I should try front facing on this flight.
I'll actually have a car in the HK end (yeah I know its a bit weird) - which is why I'm bring this with me.
Trying the seat forward-facing sounds like it might solve the issue and as you'll have a car taking the seat is certainly preferable (it's good that rental companies offer car seats, but as the history of them is not known and the child might not be comfortable in them, having your own if at all possible is generally almost always better).
It sounds like the seat you have is well-suited for traveling which is great. We found even with our daughter's Britax Frontier which was heavy and big, once we had traveled a few times and figured out how to best carry it, it became really quite easy to travel with. Obviously our daughter was at the age where she could help (she actually learned how to install the seat herself), and that's obviously different at 22 months, but there are things like small collapsible wheeled carry carts you can buy to move the seat around if that would help (we've even seen some kids at airports travel in the seats when the seats are attached to the carts).