I think the point that we can't actually know when that "last week" will be is a bit of a problem - as others have said, we do actually have IRL lives that sometimes mean we can't get to play for several days at a time, and any given player could have contributed millions worth of damage only to miss out the recognition because they couldn't get on in the last week (that no one can know in advance).
Even if it's possible to get the 2 million in only one single attack, that presumes that you're able to get on in that last week (whenever that will be) in order to do it; which just seems arbitrary - any damage done in the last week is building on damage done prior to that, so I'm not sure why singling out a seven day period counting back from the time it's destroyed makes sense.
Again, there have to be criteria, I'm not saying it should just be a free for all otherwise the entire thing is meaningless - and yes, actually playing the game and having a good time is a reward in itself, but I perceive the cosmetics as recognition for your effort rather than the reason for doing it (much like getting a medal for serving in a military campaign is recognition that you were there, not the reason for going) so it would be a shame to have put in a lot of effort only to "fall at the final fence" while others build on your work and get rewarded.
I don't know, any system is going to have downsides, but the combination of variable time frame (we don't know when the "last week" will be) and inability to actually track your progress in game/know whether you have met the criteria (the way you can with a CG using in game tools and third party tools like Inara) make it a tricky one for me to accept as being a good plan.
Only other thing I can think of would be to make killing each Titan a Community Goal somehow, which would let players see their progress and so forth. Not sure how that would work though...