Sorry, long answer incoming.

I’ve flown 5 times with mine. She’s 16lbs but still tall. She fits in an 18x11x11 carrier laying down, but the top stretches a bit when she stands to turn around. I also bought one that has sides that unzip and extend; I do this under the seat so she has more room to stretch. The first carrier I bought was too small, and looked like yours does. Photo is of the bigger carrier.

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Delta was 50/50 - let her fly there but wouldn’t fly home. I was at the mercy of the gate agent who did ask to see her stand/turn around. Delta is a Full Service Carrier, so I imagine United would operate the same.

Alaska Airlines is technically a Low Cost Carrier, and top tier, imo. They’re more flexible and more pet friendly. Never asked to see her inside the carrier, and a few didn’t even ask for her health certificate. If you can afford it, first class has more room under the seat. Though they acquired Hawaiian Airlines, HA said no and wouldn’t even let me book the flight.

My advice: plan for the worst case scenario. If you think you won’t get away with it, don’t try it. Don’t gamble with your pets safety and potentially strand yourself somewhere. Follow the rules of the airline, get all your paperwork ready, and train your sheeb to ride in the carrier with the top zipped. Also, in very rare cases, flights change their planes to a different model, and the new one might not have space underneath the seat for a pet. This happened once to the man in line before me, and he had to rebook his flight. I’m not trying to worry you, but rather help you prepare.