My late Labrador once got stuck in a doggie door, but it was not because he was too fat, it's because the doggie door was too small :~)
Poor guy, I left him at a friends house and we went out to dinner, he got his head and one arm out trying to follow her dog, and got stuck there till we got home ;~\
Awww, poor doggie. I think that if their head goes through they think the rest will follow just fine lol.