I didn't think it was much of a fightBeddoe was way better.

No, he liked and respected Beddoe Beddoe had saved his life while jogging. He was a fighter who was known for beating his opponents to death and after seeing Beddoe in the bar fight hold his own versus several guys he KNEW he'd have to kill Beddoe to win, because Beddoe wouldn't back down and he did NOT want that to happen. (This is why he begs Philo to stop after he breaks his arm).
He agrees to fight Beddoe only after they save Lynn. They fight each other out of pure curiosity of "who really would win" not for money or to kill one another.
It's a great scene shot in beautiful Jackson, Wyoming.
"I don't want your watch, man. I want your friendship!" - Lightfoot

Its a guy thing
If there be a godthan hide from him our most evil enterprise!