I can confirm that the badge is granted to SGs well after the progress bar fills to the point of appearing full on the SG badge display window. My SG has been after this badge essentially since release, since it is the only badge that we could clearly work toward - the others appear to be things that happen over time and are beyond substantial control by the player.
Please note that the item granted by this badge is next to useless; you'll receive a badge much earlier on for earning influence which provides an upgraded version of the item which is granted by this badge, thus negating the need to concentrate on CoT missions.
Another odd note here - 3-4 days after the Preorder period began, so probably around Nov. 2nd or 3rd, all the progress we made as an SG toward the Thwart the Thorns mission was completely removed, at least according to the progress bar. Our SG is Or Die Trying, on the Virtue server, and was formed during the first few hours of the preorder period if that helps anyone track down the issue. The game was so buggy at that point that there seemed little reason to report it as a bug; we were certain we would meet the badge requirements shortly and simply went on plugging away at missions. Have other SGs experienced a setback when working toward this badge?
Finally, a note to those who are attempting to get this badge as a group - so assuming 6-8 man parties. Anyone should be able to solo a CoT mission on Villainous at any level, but for groups this is simply not the case.
You'll find that you're most efficient against the CoT between level 20 and level 30. Prior to level 20 Ruin Mages and their Earthquake attack will often prove to be an unsurmountable obstacle because no villain set has support or defense as a primary skill set, so you have absolutely no way to nullify earthquake. After level 30 Succubus and Nerva Spectral Lords will utterly destroy most parties due to the fact that there is no effective defense against psionic damage which has been added to succubus, and the Nerva Spectral Lord debuff is unnecessarily powerful; even with two Accuracy SOs in your powers, a single NSL can render your accuracy totally unacceptable for combat and two of them can drop your accuracy to the point where you can not hit at all. If your healer is Kinetics or Dark Miasma, you can save him some trouble by picking up a shovel and digging your own grave.
Finally, if you get into later missions with both Wailers and Circle of Thorns, I suggest that you immediately abandon the mission until you reach level 40. The wailers resistance debuff combined with the succubus' psionic damage results in one shot kills against brutes that attempt to tank; obviously, Mastermind pets don't fare much better. If you have a large number of Stalkers or Dominators they're still doable, but debt is a major issue here and it's best to wait until debt 40 so debt doesn't matter.