While it was originally sold for around $10 on Steam, Xbox Live Arcade, and the PlayStation Network, here is the current status of how to find it:
I remember having Battle Los Angeles game when I was younger. I remember it was an FPS game and followed the movie pretty closely. Reddit·r/xbox360 Battle Los Angeles Free Download
A short tie-in to the 2011 movie where you play as Corporal Lee Imlay defending Los Angeles from an alien invasion. While it was originally sold for around $10
Was There Ever a Physical Battle: Los Angeles Game? : r/xbox360 Was There Ever a Physical Battle: Los Angeles Game
No, there is no official "free-to-play" version of this game. Because it has been delisted, it is no longer available for direct purchase or free download from its original official storefronts. Battle: Los Angeles - Steam Community