It is the Art of Sereging - circumventing the intended difficulty of challenges by following the letter of the law, but not the spirit.
Leaving out the entire rest of the game, Cyberdemon and Mastermind pose a huge problem to anyone doing this without melee. I originally attempted the successful method demonstrated in this mortem and
my own due to massive failure at the Cyberdemon on prior attempts - the amount of damage you can do with a build other then melee takes a very long time to kill the cyberdemon, and the lack of inventory for ammo is all but prohibitive towards completing the level that way. To avoid damage, which you absolutely need to do because you can't carry spare armor or medkits, you have to have Dodgemaster. This gives you basically two builds, Fireangel or Gun Kata. Gun Kata doesn't get to use its instant reloads, because you are never finishing anything off, so that eliminates half the benefit, although you still get increased fire speed every dodge. However, speed isn't really what you need here, as the issue is conserving ammo, and what you really want is damage, which only comes from the two levels of SoG in this build. SoB is prohibited so even with extra traits you've got no help there. Fireangel has something of the same issue - you reload faster, but you have nothing at all to help your damage, and SoB is once again blocked.
In both cases, you have tools to help you play safely and not take damage, but this is to some extent offset by the fact that you need to play as close to the Cyberdemon as possible to maximize damage with the shotgun or accuracy with the pistol. In the first case, you have 100 rounds available, in the second you have 200, and with the Cyberdemon's 450 health and 4 armor on Nightmare, while I haven't actually done the math here, I have serious doubts about whether either build can finish the kill before running out of ammunition.
This is, of course, assuming you arrive at Babel with 2 full stacks of ammo and nothing else, and that you make no mistakes for a very long and painstaking fight. It also assumes you killed enough original spawns to rack up the experience for a Master trait by the time you reach Babel - not really the way you want to play a Nightmare game with essentially no inventory.
I suppose it could be possible with exceptional luck with mods, exotics, and/or Uniques, and of course you could hope to get lucky with a nuke and a globe on Deimos 7, but neither of these are good strategies for a still demanding Nightmare run - any number of other factors can throw out the run, especially with a challenge like this with very little room for mistakes, and alot of reliance on luck with respect to finding the specific kind of ammunition you need(or a former that carries it and can be safely respawn farmed) at the right times, because running out of ammo almost guarantees the failure of the run(without additional exceptional luck).
Melee solves all of these problems - you can conserve your valuable ammo and so have the option of using one of your two inventory spaces for some kind of lifeline like a medkit, homing phase, or extra armor, so you can get survive more bad luck. The global speed boost the challenge gives you is a bigger advantage in melee then with guns, and also doesn't speed up the use of your ammo. Bosses become much simpler, since you can build Berserker(or hell, even Malicious Blades if you're feeling cocky) and increase both survivability and damage output, and again you can keep a lifeline item and not worry about running out of ammo or playing extremely carefully. It also means you can have your most essential traits with a much lower character level then non-melee requires.
Sure, you have the added challenge of swapping weapons and being careful with what's equipped when something dies, but it's a pretty worthwhile tradeoff as far as overall difficulty is concerned, in my opinion.
If someone were to post a mortem of this badge in the spirit of the law, not just the letter, that would very definitely be an Angelic achievement in my opinion - and, like all Angelic badges(with the exception of Shottyman, due to the [presumably] unanticipated easing of difficulty caused by several Dual Angel combinations - hey, more Sereging!), means it should be all but impossible.
P.S. Tormuse, I note that your signature indicates that you're still at 7 Angelics - in my original mortem, you mentioned taking my skepticism of the possibility of legitimately completing this badge as a challenge. I now renew that challenge, if you're feeling up to it - certainly you are among the very few people who might have some shot of actually accomplishing it. Best of luck if you choose to do so, but please, get a working keyboard first ;)