You do get to keep the badge though, right? I don't remember how it works, tbh xD
Yep, you don't have to win, just have to fulfill the stated conditions - for reference consider the bronze badges that only require reaching a certain dungeon level.
Zicher, the video is up in its entirety now. I basically phased to the hellgate(in my experience, phase devices[regular not homing - I get odd results with homing so I don't like to use them] tend to send you behind the bruiser rooms more often then not), ducked behind it and shotgunned the bruisers, then camped behind their chambers for everyone else. Tactical boots and the basic and advanced traits required for Fireangel were key. It's worth noting that with Fireangel, you can also rocketjump backwards or forwards from the ambush without taking damage if you don't find phase devices, although I never managed to finish the Anomaly without taking some damage somewhere when doing that - it's a lot more difficult then simply phasing to the back and camping.
It's possible that Anomaly is more difficult then Babel, but since I can pass it and then still screw something up in Deimos or at Babel, I've had more successes overall there, so my perception of difficulty is probably skewed by that.