The evidence that Luke might have used a priest as one of the sources for his gospel can be broken down into statistical evidence of a focus on the temple or on the priesthood, and evidence that would require the expertise or special knowledge of a priest. The evidence that he was not likely to have had assistance from a priest is to be found in source criticism.
It is proposed that Luke has a Temple focus, as one might expect if the author was being advised by a priest, so I did some elementary statistical analysis to see how strong this focus is. Of the synoptic gospels, Matthew refers to the temple 17 time, Mark refers to it 11 times and Luke refers to the temple 19 times. As Luke is noticeably longer than the other two gospels, the difference is not sufficient to suggest a focus on the temple. Far from having a temple focus, Luke 24:52-53 is a beautifully written ode to Jesus that no priest faithful to the Jewish religion would be likely to have suggested to the author of Luke or to have assisted in writing.
Many scholars have noted parallels between Luke and the works of Josephus, so if the author had a source for the story of Jesus enthralling the Jewish scholars in the temple, with his knowledge and understanding of the scriptures, some scholars see this as influenced by Josephus' account of his own childhood, with Jesus achieving the same milestone two years earlier than Josephus did:
Life 2: Moreover, when I was a child, and about fourteen years of age, I was commended by all for the love I had to learning; on which account the high priests and principal men of the city came then frequently to me together, in order to know my opinion about the accurate understanding of points of the law.
Since Josephus was a former priest, technically we could say that this is one reasonably likely case where Luke did have a priestly source. I submit that there is nothing in the other passages cited that require the author to have used a Jewish priest as a source:
- Luke 1:9 states "According to the custom of the priest's office, his
[Zacharias'] lot was to burn incense when he went into the temple of
the Lord." That a priest was assigned to burn incense would be known
to a priest, but also widely known in Jerusalem.
- It was well known to all in the first century Jewish milieu that all Jewish boys must be circumcised and that all first-born must be presented in the temple (Exodus 13:2) - see Matthew Henry's Commentary.
Finally, to source criticism. The special source (L Source) suggested in the question as one of Luke's sources is part of one hypothesis intended to help explain the synoptic problem. The focus here is on satisfactorily explaining the sources of the synoptic gospels, which clearly have a literary relationship, with nearly all New Testament scholars acknowledging that Mark's Gospel was the major source (along with 'Q' and other, special sources) used by the authors of both Matthew and Luke. At the same time, scholars say that Mark was written approximately 70 CE, which means that Luke's Gospel was written probably decades later, perhaps even in the early second century. This relationship means that, at the time Luke was written, the temple no longer existed, there were no longer any Jewish priests, and the Sadducees, who provided the temple priests, had faded from history. The author of Luke does not acknowledge receiving assistance from any Jewish priests and indeed is unlikely to have known of any. Perhaps he knew of some oral or written sources passed down from a temple priest, but I believe the examples given in the question are not strong evidence of this.