For those interested in the source, it is also known as “Dialogue with Trypho”, Justin Martyr lived roughly from the beginning of the 2nd century up until the early second half of the same century. He is one of the very earliest Christian authors.

EDIT: tl;dr: Yes I think he's blowing smoke.

EDIT: Reference to "Dialogue with Trypho" is a mistake on my end - it is an independant work in which he mentions Bethlehem as well and I guess I wasn't paying attention as much as I should have.

Heya - I'm a part time research assistant at my Universitie’s chair for Patrology and Early Church History. Perhaps I can offer something helpful, but I am aware that you are asking about the general practices of Roman record-keeping in connection with the birth/life of the Historical Jesus - I have little to offer in that regard. If the question is “would there have been records” - probably not, but I'd like to point your attention towards something else:

Jesus probably wasn't from Bethlehem. It is generally agreed in my field* that Jesus was indeed from Nazareth, early Christian sources refer to Christ as being from Nazareth, and also you might note that it is awfully convenient that Joseph would have come from a place from where the great king David had come from and also the Messiah is rumored to come from (Micah 5,1) when he finally arrives and it then just so happens that there is a census, everybody has to return to their home city for no good reason, and so on, and this in a book trying to convince its readers that Jesus is indeed the Messiah prophesied by the books of the Old Testament. Then as you already know there are problems with the dating, etc. The whole thing is, I believe the technical term is, "kinda sus". Other than that, all the references to where Jesus came from are Nazareth and Galilee. As for sources, I admit I find it amusing that this time I get to legitimately tell you to “read the bible” – specifically the gospels.

There is another Problem, and that is that it is far from proven that the Bethlehem that is attested for the centuries after Christ is indeed the Bethlehem mentioned in the old Testament (which is also attested in the Amarna letters). There is a fair chance that that what is today Bethlehem was only ascribed as such during the period in which also Justin Martyr writes (first half and middle of the second Century). He is indeed a great example of this aspect of early Christian literature, which is trying to prove that Jesus was the Messiah (or "the Christ", if you will, as that is what it means). Luke is a great example of this as well: On a side note, if the Marcion hypothesis holds true it would make the author of the Gospel of Luke and Justin Marty roughly contemporary, and this theory has been gaining a lot of traction these days and is also something I am currently involved in (as an assistant, mind you, you won’t be seeing my name on any publication on this anytime soon). As to why this is likely, firstly there are countless historical examples of where the fulfilment of some prophecy or writing was ascribed retroactively, and second is that there is just no proof - in 2012, Ely Shukron of the Israely Antiquity Authority claims to have found a seal proofing that the contemporary Bethlehem is the Bethlehem of the Old Testament, but to my knowledge, he has yet to publish his findings. If you would call this splitting hairs, you’d be right, but I’m adding this for the sake of thoroughness. It is perfectly possible that this is the Bethlehem of the Old Testament, however, this has yet to be proven, and there are legitimate doubts as to whether it was inhabited during the time of Jesus’ life (The area itself I believe has been sporadically inhabited since the neolithic age).

In closing, I’d like to point out that it’d be a bit odd if there were any continual record keeping from before and after the first Jewish-Roman war, but that’s just a thought and something I have nothing to offer for in terms of arguments. I hope this has helped you in some way, or if not, at least given you food for thought. Perhaps someone else will contribute some more, I’m always amazed with how some people go above and beyond with their replies, but if it can at all be avoided I’d rather not write full-blown papers for Reddit.

*There are, of course, still a bunch of diehard fundamentalists that believe the Bible is an infallible history book. I guess it's kinda in the Nature of the thing. I know plenty of believing Christian scholars that go with the historical-critical method.

I think it would be helpful to start with an introduction to how we can interpret the historicity of stories like the two different nativity stories in the Gospel narratives of Luke and Matthew.

It would be unimaginably great if we had contemporary accounts by perfectly interested but uninvested observers to learn about the life of Jesus from, or better yet multiple independent ones, but the contents of the bible really are pretty much the best we've got for figuring out what actually happened. It was formulated by committee in the fifth century, but that committee did a remarkably good job with the remarkably decent materials they had. The accounts we have are written by true believers, who were not themselves eyewitnesses, and who were writing in a different language and living in a different place than the eyewitnesses. They are also not free from collaboration (With Mark being used as a source for both Matthew and Luke), and particularly in the Nativity story they can be pretty wildly inconsistent in both details and global understandings.

However, there is still a lot we can do to come to remarkably solid conclusions out of what we've got. Thankfully there is a common thread among an extended community of puzzle solving oriented people who have obsessed about these kinds of questions for centuries. Since well before the enlightenment, people have been putting a lot of thought into squeezing just about everything that we possibly can out of the extant records we have. They've found that when assessing the veracity of historical materiel, it is important to keep in mind a few more principles, not all of which are very intuitive,

• First, and intuitively, the earlier the sources that the material is found in the better. Even just twenty years can be an awfully long time to be playing a game of telephone, or even for a single person to keep a consistent view of something. We do have pretty reasonable ways to date even the earliest texts, for example each of the gospels refer to the destruction of Jerusalem (even if it is sometimes as an awfully specific prediction) and so we can reasonably assume that they were each written after that.
• Second is the criterion of embarrassment. There are a bunch of parts of the New Testament that really don't fit in the simplistic version of the Christian narrative, and these are, if anything, parts that we can trust the most. Why would anyone make them up later? In a lot of first and second-hand accounts in ancient texts, and including the bible, you will often find things that just make too little sense to be fiction. For example, during Mark's very condensed account of the final arrest of Jesus in the garden of Gethsemane, Judas kisses Jesus, the Romans show up, Jesus gets sarcastic, and everyone but Jesus books it, but then something really interesting happens. An apparently random unnamed dude, it's not even clear if he was a follower of Jesus, loses his clothes as he tries to flee naked. Our immortalized streaker adds absolutely nothing to the story, isn't the least bit relevant to the narrative, and if anything detracts from the message the author of Mark is trying to convey; but heck would that be memorable to an eyewitness. In a time when to be naked was to be dishonored, and to be dishonored was to be less than human in a way that is only really understandable in the abstract in today's world, that was a pretty big deal. While it would never occur to a fiction writer to include this, an eyewitness talking to the author of Mark would have good reason to consider the tale incomplete without it.
• Third is the criterion of multiple attestation, or the more sources we have that cite or repeat the material the better. Material found in multiple sources that are independent of and contemporary to each other is more likely to be historically accurate. It is pretty intuitive that it would be difficult for someone to make something up and get someone else, somewhere else, to make up a similar thing at the same time. Thus many authors saying something in 75 CE isn't necessarily that much worse than one author saying the same thing in 50 CE. For example, both Matthew and Luke talk about how Jesus is from Nazareth but say very different and unique things about how he got there from Bethlehem. Mark also says that Jesus was from Nazareth and so does John, which was written independently of the other three Synoptic gospels. Thus, we can pretty solidly trust that Jesus was from Nazareth. However, as we can assume that since both Matthew and Luke were aware of the prophecies that suggested that the messiah would be born in Bethlehem, their very unique stories of the nativity are probably a result of their common need to explain how Jesus was both born in Bethlehem and famously from Nazareth. The traditional Christmas stories that many of us get as children are generally either one, the other, or a pretty forced mash-up of the two. With this in mind, we can also trust that Jesus did indeed come from Nazareth all the more using the criterion of embarrassment. Nazareth was a two horse town in the middle of nowhere that was famous for precisely nothing and recognizable to practically no one. Particularly when Bethlehem, the birthplace of David, would make a much more reasonable origin for the Messiah as the author of Matthew explicitly notes by quoting prophecy in Micah, why make that up? Even so, how could you possibly get everyone to agree on it if you did?
• Fourth, is perhaps the strongest, basic coherence and just making sense in context. Jesus was an itinerant rabbi in the first century Levant, and any traditions that don't make sense in that context are a lot less reliable. A lot of the later non-canonical Gospels contain things that are pretty wild, but even some of the canonical gospels have some subtle things that don't make sense when you think about them hard enough. For example, in John's account of Jesus' famous late night conversation with Nicodemous, Jesus tells him that he must be born again/above. It is a play on words in Koine Greek, and kind of a neat one. The words used are gennao (Strong's 1080), which means begotten or born in a formal father oriented sense, and it is modified by anothen (Strong's 509), which can mean either again or from above. The author of John uses anothen for both meanings in different parts of the Gospel and so the effect is obviously intentional. However, importantly, while it would have been absurd for Jesus to have been speaking Greek to a Pharisee like Nicodemus, neither the Arahmaic nor Hebrew languages that Jesus could have been speaking have an analogous word with both meanings.