While I can’t really discuss how or if Constantine himself influenced modern Catholicism, but I can address some of the other points that you made in your post. I actually focused on Christianity in late antiquity, especially during the reign of Constantine, for a chapter of my MA thesis, which is where a lot of this information will be coming from. So with that out of the way lets dive in.

So the first point that I briefly want to touch on is that you seem to be implying that Constantine was not a “real” (for lack of a better term) Christian because he was not baptized until he was on deathbed. But it should be noted that Constantine seemingly believed that if he was baptized any earlier then he risked not getting into heaven because of any sins that he may have committed after the baptism. So instead of being baptized upon his conversion, or shortly thereafter, he instead remained a catechumen, until being baptized just before his death. Now, one could argue that Constantine made this decision as a sort of loophole so he could continue to do things that would have been viewed as sins by the Church, but that’s just speculation and even if it were the case it doesn’t really invalidate his status as a Christian. What matters is if Constantine himself believed in God and believed that he was indeed a Christian, and we really don’t have a lot of reasons to think otherwise.

Now as you pointed out, yes Constantine continued to use pagan imagery throughout his reign and was very clearly influenced by his past belief in the traditional Roman gods, but really that’s to be expected of someone who spent decades worshipping one set of gods only to convert to a monotheistic religion as an adult. Nevertheless, this is where things get a little tricky, and is why some people believe that Constantine was either not actually a “true” Christian, or that he was but only for political purposes.

First, I want to point out that Constantine’s relationship with Christianity was undoubtedly complex, especially in the years immediately after he converted. In the years before Constantine converted Christians had been persecuted throughout the Empire by the Tetrarchs. In fact, Galerius’ edict of toleration had only been issued the previous year following eight years of persecution (which itself was preceded by another persecution in 299). In addition, the majority of Christians at this time lived in the east, there were some Christian communities in the west but not nearly as many. So, when Constantine converted in 312 the status of Christianity in the Empire was still tenuous at best, and in the western half of the Empire Christianity was still technically illegal. That wouldn’t change until Constantine and Licinius issued the Edict of Milan in 313. Now I’m sure based on this information alone you can already begin to see why Constantine’s early relationship with Christianity was far from simple, he converted in a world where Christians had been persecuted on and off for centuries and in a region where the Christian population was still sparse.

As for the pagan imagery that Constantine continued to use during his reign, well that can be explained, at least partially, by the aforementioned circumstances. In the past some scholars have argued that Constantine’s conversion was anything but genuine, and that he only converted out of self-interest, largely because he still used pagan imagery and didn’t convert until his deathbed. But historians like Roger Collins use this as evidence for why his conversion actually was genuine. As we’ve established, when Constantine converted the vast majority of Romans still followed the traditional religion, had spent the previous eight years persecuting the Christians, and Constantine himself had spent his entire life up to that point as a pagan. Constantine wouldn’t even become emperor in the east, where most Christians lived, for another twelve years. In the west meanwhile, most high-ranking officials in the imperial hierarchy still followed the traditional religion of Roman, this includes practically all of the high-ranking military officials. So someone in a position of power like Constantine would have had to remain lenient towards pagans out of necessity in order to maintain keep his power, especially since the Roman military already had a well-established habit of overthrowing emperors that they did not like. Now, this is not to say that Constantine’s sympathy towards pagans was completely fake, its entirely believable that he still held onto some pagan beliefs himself. (Roger Collins, Early Medieval Europe)

This explains why Constantine continued to include pagan imagery on his coinage, associate himself with Sol Invictus, and allowed pagans to construct temples and make sacrifices to the gods, at least until roughly 325 when he had a more secure hold on his power in the east. His continued association with Sol Invictus is perhaps the easiest aspect to understand because the cult of Sol Invictus was rather widespread in the military. As a quick side note, some scholars have argued that since Sol Invictus was a singular deity, rather than part of a pantheon of gods, that it may have been more palatable for Christians. I don’t know if that’s an argument that I would make, especially as I haven’t done very much research into the cult of Sol Invictus, but nevertheless. Constantine’s coins even said “Soli Invicto Comiti” which basically means that the unconquered sun is a companion of the Emperor, and he in 321 he declared dies Solis as the Roman day of rest. This is especially interesting because Christians had already begun to treat Sunday as their day of rest rather than Saturday, and this may have been an attempt by Constantine to further reconcile Sol Invictus with Christianity (and perhaps to move pagans in the direction of Christianity).

Constantine’s willingness to associate himself with Sol Invictus could also have stemmed from his earlier associations with Apollo. In fact, two years before his alleged vision before the Battle of the Milvian Bridge, Constantine had a rather similar vision of Apollo handing him three wreaths that represented thirty years of power (which Eusebius later reconciled by saying that it was actually God that Constantine saw in this vision). Apollo was of course also a sun god like Sol Invictus, granted he had plenty of other roles as well. But unlike Sol Invictus, Apollo was demonstrably part of a pantheon of gods and could not be easily separated from the other gods, whereas Sol did not have that baggage attached.