My dad's last hunting rifle was a BAR chambered for .30-06. Part of the reason he got it was memories of the military BAR he carried across China, Burma and India in War2.
While it seemed heavy to me he never complained about it. It's a solidly built rifle. Shot whatever ammunition he fed it and I don't remember any problems.
He said it was a sweet shooter so I guess the recoil was to his liking. He bought it right after he took one of my 7mm's on a deer hunt. My hunting rifles are light and I load them heavy. The first couple of shots he fired bruised his shoulder, skinned his trigger finger and busted his eyebrow. And he was bigger than me.
Anyway he came back from the sporting goods store with the .30-06 BAR and never shot anything else. He left long shots to the younger guys but he got his deer every year.
I'd say if .338 is what you want go for it. You can't take it to Africa or some other foreign places that ban semi-autos but for the largest North America big game it's fine.