It's almost certainly the other way around since Battle Royale was published in 1999 and the Battle Royale film came out in 2000. Both probably take some inspiration from Lord of the Flies
Internet was hardly available in the year 2000. It takes a lot of bandwidth (and probably CPU too) to run a 100 players FPS. It wouldn't do well before broadband got ubiquitous, almost a decade later.
That doesn’t change the fact that BR predates the hunger games (though the author claims to not have heard about BR when she wrote the hunger games).
BR is supposedly about how the old of japan are metaphorically sacrificing the youth. The hunger games was (according to an interview) about reality tv meets the war in Iraq.