Does this story appear in Roman folklore?
Firstly, we should verify that the Magnus Maximus in the story is, in fact, the same Magnus Maximus as we see in recorded history.
This excerpt is from De Excidio et Conquestu Britanniae, a historical document from the 6th century AD (emphasis mine):
- At length also, as thickets of tyrants were growing up and bursting forth soon into an immense forest, the island retained the Roman name, but not the morals and law; nay rather, casting forth a shoot of its own planting, it sends out Maximus to the two Gauls, accompanied by a great crowd of followers, with an emperor's ensigns in addition, which he never worthily bore nor legitimately, but as one elected after the manner of a tyrant and amid a turbulent soldiery. This man, through cunning art rather than by valour, first attaches to his guilty rule certain neighbouring countries or provinces against the Roman power, by nets of perjury and falsehood. He then extends one wing to Spain, the other to Italy, fixing the throne of his iniquitous empire at Trier, and raged with such madness against his lords that he drove two legitimate emperors, the one from Rome, the other from a most pious life. Though fortified by hazardous deeds of so dangerous a character, it was not long ere he lost his accursed head at Aquileia: he who had in a way cut off the crowned heads of the empire of the whole world.
Picts and Scots.
- After this, Britain is robbed of all her armed soldiery, of her military supplies, of her rulers, cruel though they were, and of her vigorous youth who followed the footsteps of the above-mentioned tyrant and never returned. Completely ignorant of the practice of war, she is, for the first time, open to be trampled upon by two foreign tribes of extreme cruelty, the Scots from the north-west, the Picts from the north; and for many years continues stunned and groaning.
Source: De Excidio et Conquestu Britanniae, St. Gildas, 13-14
This sounds like the same story as The Dream of Macsen Wledig (emphasis mine):
And immediately the emperor set forth with his army. And these men were his guides. Towards the Island of Britain they went over the sea and the deep. And he conquered the Island from Beli the son of Manogan, and his sons, and drove them to the sea, and went forward even unto Arvon. And the emperor knew the land when he saw it. And when he beheld the castle of Aber Sain, "Look yonder," said he, "there is the castle wherein I saw the damsel whom I best love." And he went forward into the castle and into the hall, and there he saw Kynan the son of Eudav, and Adeon the son of Eudav, playing at chess. And he saw Eudav the son of Caradawc, sitting on a chair of ivory carving chessmen. And the maiden whom he had beheld in his sleep, he saw sitting on a chair of gold. "Empress of Rome," said he, "all hail!" And the emperor threw his arms about her neck; and that night she became his bride.
Seven years did the emperor tarry in this Island. Now, at that time, the men of Rome had a custom, that whatsoever emperor should remain in other lands more than seven years, should remain to his own overthrow, and should never return to Rome again.
So they made a new emperor. And this one wrote a letter of threat to Maxen. There was nought in the letter but only this. "If thou comest, and if thou ever comest to Rome." And even unto Caerlleon came this letter to Maxen, and these tidings. Then sent he a letter to the man who styled himself emperor in Rome. There was nought in that letter also but only this. "If I come to Rome, and if I come."
And thereupon Maxen set forth towards Rome with his army, and vanquished France and Burgundy, and every land on the way, and sat down before the city of Rome.
Source: The Dream of Macscen Wledig
Now that we know the historical figure is the same as the story's figure, we can talk a little about the history. Magnus Maximus was the emperor of the Western Roman Empire from 383-388 AD. The empire fell approximately 90-120 years later. For comparison, calling this story part of Roman Mythology would be a little bit like regarding the Spanish-American War as part of "American Mythology".
Most of Roman folklore and mythology comes from stories like Ovid's Metamorphoses - 8 AD, Virgil's Aeneid - ~25 BC, and Livy's Ab Urbe Condita Libri - ~25 BC. Note that this is hardly a comprehensive list.
However, Magnus Maximus was too close to the fall of the Roman empire to be reasonably included in it's mythos; he was emperor after the Empire split in two. This is why it is a relevant myth for the 5th century newly rising Anglo-Saxons, but not for the Romans themselves.
Note: I have decided that Wikipedia is a sufficient source for this part of answer as the facts contained herein are basic, widely accepted historic facts of the era.