The issue is just with Korean (KE) - they generally don't allow pets in the cabin on international flights done with a 737. I brought a dog from HKT to Europe in the cabin on Korean this spring with no issues. But HKT gets served with an A330, while PNH is that no-pets 737...
I would go with Asiana (OZ) - they do not have such a restriction (at least not one that is publicly available), and you should be able to take your cat in the cabin to PNH.
To my knowledge, the Koreans (KE and OZ) are the only airlines allowing pets in the cabin going transpacific (5 kg weight limit in both cases).
Second option, if Asiana doesn't accept the cat in the cabin as well: Fly with the cat in the cabin to ICN on either KE or OZ, and put it in the hold only for the five-and-a-half-hour segment to PNH from there. Depending on whom you encounter at the airport in Seoul, this necessitates entering the country and you'd need to adhere to
Korean animal import regulations (rabies vaccination and titer test, microchip, health cert).
The closest destination that gets a bigger aircraft on Korean is BKK. You will need to enter Thailand with your cat, which is a bit more complicated than transiting Korea.
And no need to travel overland - both Thai and Bangkok Airways will fly your cat in the hold for reasonable fees on that one-hour flight from BKK to PNH.
Putting the animal in the hold all the way, just checking it through - I would only do that if you know your animal(s) can handle it. LAX-ICN is 13.5 hours, plus 5.5 hours for ICN-PNH, plus the 55-105 minute layover, plus the ~two hours from check-in to departure and landing to passing customs is about 22 hours your cat will be confined to its crate with no food or water or option to relieve itself. This is the same with pretty much all other possible itineraries - total travel time is just too long.