I once heard an anecdote about a colleague who (I think) said that Constantine’s revisions of the imperial postal system were more significant than his conversion to Christianity. This may, in fact, be true, depending on how you define your terms. However, it is the case that, overall, coming to grips with Christianity will help you understand late antiquity better than knowing the imperial postal system.
If you begin with the Tetrarchy and Diocletian, you will need to have some grasp of who Christians are and why the Roman government disliked them for understanding the persecution.
If you begin earlier with the Third Century Crisis and are interested in Latin literature, the fact that we have so little Latin literature from the second century will throw you into the arms of Cyprian of Carthage and his letters.
Beginning with Constantine there is a conversion of the upper classes, and these are the people who produce or for whom are produced most of the stuff that survives from antiquity — fancy houses, poems, philosophical treatises. Their religion is thus not inconsequential. And they eventually do become Christians — we can learn about the last pagans of Rome (to cite the title of a book by Alan Cameron)
And if you are interested in Later Latin Literature, Christianity is all over the place. Some of the greatest poets of Late Antiquity write explicitly religious poetry. It would be a shame to study the world of late antiquity (to cite the title of a Peter Brown book) and miss out on Prudentius and the other Christian epicists. Likewise the Greek verse of Gregory of Nazianzus, or the sublime Syriac poetry of Ephrem and his luminous eye (to cite a Sebastian Brock title).
While the rise of western Christendom (to cite Peter Brown again) is a major feature of the study of the Mediterranean world in Late Antiquity (Averil Cameron this time), I admit one should be perspicacious. There is a lot to grapple with.
Consider the realm of texts: Augustine of Hippo is the ancient Latin author with the largest surviving corpus, for one thing. We have more Christian letter collections from Late Antiquity than the non-Christian ones from preceding centuries. Indeed, Christians love books — sermons, letters, poems, long theological tractates, canon law documents, apologies, polemics, biographies, hagiographies, liturgies, and so forth, flow forth in abundance in Late Antiquity in Latin, Greek, Syriac, and Coptic.
Material culture is also a big realm, from Spain and even Britain in the West to Mesopotamia in the East, the Roman Empire and its Persian neighbour has its fair share of physical remains, some of them the large, mosaic-encrusted churches of Ravenna, others the foundations of churches in Salamis on Cyprus. This is not to mention the myriad smaller objects of Christian origin — ivories, icons, Bibles, Bible covers, communion vessels, etc.
Moreover, Christianity is a complex phenomenon. Are we looking at the beliefs and writings and practices of the educated elite? What about the urban poor? What about different modes of belief amongst different Christian bodies? Bishops? Laypeople? Rome? Antioch? Nisibis?
In fact, there’s so much, whether you like Christianity or not, how could you help but take an interest in it if you’re interested in Late Antiquity?