GTA V Beach Bum DLC won't show up in game on Xbox 360
Actually the Beach Bum DLC is available offline as well.
According to a post it is most likely license issues with the xbox 360 (if you have a new console since you bought GTA V).
My guess is that, like the majority of 360 owners, this isn't your first 360. Your license is still registered with the original 360 unit that you created your account on.
Make sure your 360 is online.
Access Account Management (last page of the XBOX guide menu).
Access License Transfer and activate it.
Delete and re-download any DLC media that you wish to access offline (you won't have to re-purchase your DLC).