Can't edit my previous post but wanted to provide an update.
We traveled from Manila to Korea on Korean Air but had been unable to check-in online. Had to be done at the airport. Korean Air gave us no trouble at the airport. Only needed GC and extension letter. They printed our boarding pass for the flight from Korea to the US as well. Flight from Korea to US was on Delta. No issues anywhere. My wife had to scan her PH passport in Korea, and show her GC and letter to immigration here in the US. I'm not sure if it matters but our flight from Korea to the US was direct to our home state.