The Emperor Constantine decreed Sunday to be the Roman day of rest on March 7th 321, before that it was the day of the sun in the original 8 day Roman nundial cycle and in the 7 day week introduced by Emperor Augustus in the first century (which was not widely observed as the nundial cycle persisted for at least a few more centuries according to Valentinius’ Calendar of 354) but as far as I am aware it had no special meaning connoting rest or worship, outside of the sun cult within Roman religion. The only justification that I can find is from Saint Jerome, who said “If pagans call the Lord's Day the 'day of the sun,' we willingly agree, for today the light of the world is raised, today is revealed the sun of justice with healing in his rays” but Jerome was born 26 years after Constantine declared Sunday to be the day of rest, so what was Constantine’s reasoning for declaring the pagan day of the sun to be the day of rest, considering that he was a Christian? In Constantine’s declaration of the new day of rest he said “On the venerable Day of the Sun let the magistrates and people residing in cities rest, and let all workshops be closed. In the country, however, persons engaged in agriculture may freely and lawfully continue their pursuits; because it often happens that another day is not so suitable for grain-sowing or vine-planting; lest by neglecting the proper moment for such operations the bounty of heaven should be lost.” But that does not give much justification at all. Furthermore, why did the Council of Laodicea in 363 explicitly prohibit the observance of the Saturday Sabbath? Where did the importance of Sunday to Christians first arise from?
My source is Philip Schaff’s History of the Christian Church: Vol. II along with the works of Saint Jerome, the Calendar of 354 and the Canons of the Council of Laodicea themselves. Please correct me if I’ve gotten anything wrong.
To really understand why Constantine made this decision, which of course will never be proven definitively one way or the other, one must first understnd Constantine's relationship with religion. Firstly it should be stated that Constantine's claimed vision of the cross and the message "in hoc signo vinces" was, of course, more than likely a fabrication on his part to ingratiate himself with the Empire's Christian population. This wasn't even the first time that Constantine had used the claim of a religious vision to cement his right rule:
For you saw, I believe, Constantine, your ovm Apollo with Victory accompanying him, offering you crowns of laurel, each one of which brings you an omen of thirty years. This is the number of human generations which are owed to you in any case, beyond even the lifespan of Nestor. (5) But why do I say 11 J believe? You saw him and recognized yourself· in the appearance of that god, to whom poets• divine songs prophesied that sovereignty over the entire earth was due. ( Panegyrici Latini VI)
Of course this claimed vision, from 310, was distinctly pagan and Constantine claimed that his right to rule came directly from the gods Apollo and Victory, the poem also goes on to compare Constantine's physical appearance and characteristics to those of Apollo (Constantine quite literally sees himself in Apollo). So at the very least Constantine was not unpracticed in the use of divine "visions" to legitimize his rule of the Roman Empire. This claimed vision of Apollo is also important for another reason, because the Romans had come to associate Apollo with the sun god Sol Invictus (the unconquered sun). Sol Invictus may have also been viewed more favorably by Roman Christians than the other "more" pagan gods like Jupiter and Mars.
Now as for Constantine's actual vision before the Battle of the Milvian Bridge, after which he allegedly had his soldiers put the symbol of Christ on their shields (the Greek letters chi and rho). While there were some coins, and a medallion, issuded between 315-318 with this symbol it was actually quite rare in imperial iconography before the 320s. So even if Constantine did first begin to subscribe to some small portion of Christianity immediately before or after the battle, it was not important enough to have a significant place in imperial iconography for several years.
It should also be noted that Constantine claimed Sol Invictus as his patron god after 310, before 310 he claimed the god of war Mars. Claiming Sol Invictus as his patron served several purposes for Constantine: it emphasized his status as his father's heir , his devotion to Apollo, and by extension Augustus himself as Apollo was his patron god, and it also was less offensive to Christians because the worship of Sol Invictus was more monotheistic in nature than the traditional Roman pantheon. The cult of Sol Invictus was also very important to Constantine, who took several measures to ensure that he would be associated with the unconquered sun. The Arch of Constantine was built in 315, and while its only religious inscription simply refers to the "divine" (possibly as a way to please anyone who read it no matter their religious beliefs) it was built so that anyone approaching the Arch from its main path would be able to easily see the colossal statue of Sol Invictus (which seems to have pretty obvious implications). So while Constantine undoubtedly began to support the Christian church in 312, conveniently after his victory over Maxentius, this could have been done for any number of reasons, and Sol Invictus undoubtedly remained important to him until at least 325. In 311, shortly before he died, Galerius had ordered an end to the persecution of Christians that had been started by Diocletian, and Maximinus II was the only one who refused to obey. So by 312 Christianity had been "tolerated", to some degree, for a whole year which likely influenced Constantine's own tolerance of Christians.
As you noted, Constantine declared the day of the sun as the Roman day of rest in 321, four years before Sol Invictus no longer appeared on his coins. So it's possible that Constantine actually wanted to establish a connection between the Christian God and Sol Invictus, perhaps as a way to ingratiate the still substantial pagan population with the growing Christian population. In 323 Constantine even passed an edict which banned Christians from taking part in the state sacrifices:
Clergy shall not be forced into participating in pagan practices; anyone who forces a clergyman into such an act may be fined or publicly beaten, depending on his legal status. (Codex Theodosianus 16.2.5)
There are several other laws that Constantine issued to protect Christians as well, for instance reserving certain religious privileges for Christians, Jews were banned from stoning Jewish converts to Christianity, Jews were banned from owning circumcised slaves, etc. So it does seem that at the very least Constantine did want Christians and Pagans to coexist peacefully within the Empire, even if his own status as a Christian wasn't as legitimate as he wanted it to seem. Some historians, like Patricia Southern, go so far as to argue that the fact that Constantine did not change his mind on Christianity and outlaw it, which Southern says he likely could have done with relative ease, indicates that the Church most likely gave him what he wanted. Constantine also limited the number of people who could enter the clergy at any one time, because it was used as way to escape their "municipal burdens". So it would seem that Constantine most certainly used Christianity to serve his purposes when it was advantageous, but also had absolutely no problem in curtailing it when it was advantageous to do so. So while there is not, and probably cannot be, one single answer as to why Constantine made Sunday the day of rest, it probably served several different purposes at once, and was influenced by several factors. I am also by no means an expert on early Christianity, so I welcome anyone who wishes to expand upon this from a more Christian point of view.
Sources:
Timothy David Barnes, Constantine and Eusebius
Patricia Southern, The Roman Empire from Severus to Constantine
Panegyrici Latini VI
Codex Theodosianus