Dude it's one of the most famous songs ever off one of the best albums of all time, Bioshock did not need to endorse it for it to get acclaim from the BBC. It was to promote a new channel launch, and the involvement of Brian Wilson (who will be a hero to most of the people involved) must have been a big draw for the talent.
Well felixjmorgan here is how I came to guess, that they took inspiration from Bioshock Infinite.
The release is little over a year past the time Bioshock Infinite was released so their is a certain proximity to its release. In the game the song stood out to most players because the setting is 1912, 54 years before its release, albeit sung as a Barbershop Quartet. Also when searching for this song, It's one of the highest results beyond the original bands performance and this BBC cover.
In the music video at 1:44 you see a woman in a giant bird cage. A motif used in the game.
Several woman have wings another motif used to reference the other main character Elizabeth in the game.
The end of the video is played among the clouds, just as it is in the game.
Now while maybe, they came up with all of this on their own and that these similarities are unlikely, but go fuck yourself.