I just returned to the US from international travel. I already have my physical biometric waiver letter but didn't think to take it with me so I left it at home and I was fine without it. I showed my expired green card, the extension letter and my passport.
I recommend you allow some extra time for them to review these as I couldn't check in for my flight online** and almost missed my flight out of New York. I had to wait in a long line to get my boarding pass - the airline rep was very confused by the paper extension letter so I had to wait for his manager to come and look at it - cue round three of a long wait. It seemed they hadn't seen this piece of paper/extension letter before (they were holding it up to the light etc.). Not a huge deal as I guess we're more concerned about being allowed back into the US!
When I flew back to New York yesterday, it was a breeze (I have global entry so not sure how that effects things). Side note, the immigration officer at JFK looked at my extension letter and said "24 months?! It used to be 12 to process! What a joke, what are they doing over there?!" *preach*
**On my return journey, I decided to try check-in again online via the airline app and it dawned on me that in the green card expiration date part of the check-in form, I could technically put 2024 as the year of expiration and voila, it worked!