So I was thinking…
They should ground the new competitive playlist in the fiction! Wtih that in mind, “Team Throwdown” is kinda…well…it’s gives no sense to the “premiere competitive playlist” idea.
In order to ground it in the fiction, they should,
A) Call it Spartan League (gives a “sense” to some type of “official” league) People will start referencing it a lot easier that way. This will create a lot of social and media buzz “Oh what’s your rank in Spartan League?! Mine’s 45!” ect. #spartanleague 
B) Once the CSR system comes out create a cinematic where sarah palmer give a prep talk and introduces us to the it all(both the playlist and the CSR system). Kinda like when bungie did the whole reach beta thing with aisha tyler.
C) Offer prizes (both large and small) and tournaments on a semi-regular basis.
As for the other names that have been proposed…
Team Throwdown gives no sense to the “premiere competitive playlist” idea and Team Hardcore will only intimidate most people from playing it. We need something like this to keep peoples attention and coming back for more. The name Team Throwdown won’t do that and neither will Team Hardcore.
Spartan League is something we can all relate too. Both Hardcore and Casual alike.