We were just over 5kg with our pup's cage a few years back, and we weren't able to book the ticket after mentioning that. We didn't try to push it at the check-in, but I think it's a safe bet to say that it's pretty firm. Our dog and his cage could certainly have fit under the seat.
But flying with the dog in checked baggage on Korean is fine. We've had great support from Korean each time, from the flight crew confirming that he's made it on board to ground crew confirming that he's in the layover area of the airport and doing fine.