The lack of a regular QS place at BW is my only complaint about that resort. I've been to Big River once and it was probably one of the worst meals I've ever had in Disney. Food was okay, nothing special, but the inside of that place was so gross. Very dated, things were just not clean, layout was so weird...you can really tell it's run by a third party.
A&C is a good suggestion, but you probably don't want to walk all the way over there while you're trying to get settled in the room and everything. Have you checked the menu at the pool bar, Leaping Libations? They're open late and don't have a burger specifically, but they have hot dogs and stuff. Or just order DoorDash or something when you're on your way to BW, have dinner delivered around the same time you arrive.