I stumbled across a solution to this achievement that admittedly won't work for anybody who has already started the game, but I hope it can help people who are reading this before they begin.
After turning on the game for the very first time, my friends immediately invited me into a public multiplayer Halo 2 match. As soon as the match was over, this achievement popped for me, making it one of my first three achievements in the entire MCC.
It seems that, perhaps, launching straight into a multiplayer match before doing anything else in the game bypassed the start of the achievement's three-month timer. When I say that I didn't do "anything" beforehand, that included not customizing my profile or even changing my game settings. I simply turned on the game for the first time, entered a multiplayer match, and watched the achievement pop.
For the record, this occurred on April 9, 2015. Please leave a comment if this solution worked for you, because I fully admit that I'm writing this based on personal experience and so I don't know if the achievement bugged out in my favor or if this is a real workaround to earning the achievement without a wait. I'll happily edit (or delete, if need be) this solution if it doesn't work as explained above. :)