First, welcome aboard!
I've read the parts of Gibbons work that pertained to Emperor Constantine, in an effort to build a picture of the consolidation and legalization of Christianity. I have only general information prior to his reign committed to memory, and less after, apart from understanding that "Rome" didn't completely fall until the Ottoman Turks overran Constantinople sometime in the 15th century I believe it was...in other words, the Barbarian sack of Rome didn't end the Empire, only the western half of it.
As with all of history, I think the causes are not as simple as "this" or "that." Both played a role. I suspect there was continued infighting among various factions vying for power that weakened the Empire from within, and ineffectual defense against an outside aggressor who was able to capitalize on the internal chaos. That would be my unstudied guess on the matter.
While I have heard the flippant argument of Christianity being the undoing of Rome before, I think it is relevant to remember that when Rome had Christian administrators, society on the whole functioned much more smoothly. Christians could read, typically, which made them better suited to administration...even before the Persecutions of Diocletian and the edicts of toleration. During the persecution it was learned that when literate administrators were replaced with illiterate administrators, a lot of the social aspects of governance tended to unravel. So it is a bit of a stretch to my understanding how "effective" administrators could be directly responsible for the implosion of the government, not without some serious conspiracy and collusion of which there is nothing to indicate. That there were vying factions competing for power in the government isn't any surprise...Rome had always had vying factions competing for power throughout its history, is was simply timing and coincidence that some of those factions happened to be affiliated with one or another branch of Christianity. So depending on bias and political spin, it could be made to appear Christianity was the culprit...which is a position I do not believe to be correct, as that would be a misapplication of the facts to support a bias...something altogether too common among western scholars of the past 200 years in pushing an atheist agenda.