the version without the flat note is known as "the Monastic Salve" commonly chanted in Benedictine, Cistercian, and Dominican Communities with some differences occasionally (my own community sings it as well)

While the second version with the flat is known as the "Solemn Salve" and is the version more sung among Clerics, or the Roman version if you want to call it so.

So, neither is technically incorrect.
I have a folder with about 100 Salve Regina Tones, that have variations from different countries, regions, and congregations. When we understand the history or this chant and how a lot of it was passed down not necessarily by sheet music, you can understand why variations occurred.

My personal melodic instincts - FWIW, which is nominal at best - are for TI if the line continues upward, and a TA if continues downward - because most amateur singers tend to shade pitch that way anyway (and, if that's a typical tendency, it's best to turn it into a praxis - even if merely by notation - that everyone can be directed to do together instead of individual hit-or-miss; this falls adjacent in my mind to the practical rule that the way to pronounce a word is whatever the music director tells you to do).

The question about THE correct version is ill-posed in the context of oral traditions. In this case it is even more so as your question is about musica ficta. See here for a gentle introduction.

In this case of a turning note A-B-A, one might paraphrase an ironic shirt by Early Music Sources: It is alwas si bemolle (B flat), except when it's not ;-)

In this particular case, I would strongly prefer si bemolle because there is a prominent turning note F shortly afterwards, which would create an awkward tritone with a si naturale.

Note that there is even another option for this frequent melodic figure (ex. "Gaudeamus"), i.e., to replace the si (B) with an ut (c). I have frequently (?) observed this in Renaissance cantus firmus settings, for example in Isaac's famous setting of "Virgo prudentissima" where he uses it for the phrase "quasi aurora". I doubt that this was his deliberate choice, and guess that this was the actual plainchant version sung at Isaac's time at the court of Maximilian I.

ha.

OTOH I only think that the tritone matters in the passage from F to B/B to F directly. It is much less dissonant when you are simply descending down the scale diatonically.

Well, the Si/B/Ti to Do is very much what happened "naturally" as we see in mode 3's associated psalm tones (also mode 8, but for whatever reason, only restoring the dominant of the former ever got any traction).

The question of a "correct version" is ill-defined for a lot of medieval pieces that were transmitted as part of an oral tradition. In this particular case, there is the additional problem of musica ficta. See here for a gentle introduction.

To paraphrase an ironic shirt by Early Music Sources, one might say that in this case (upper turning note above la) it should alwas be a si bemolle, except when it's not ;-)

In this quite common musical figure ("gaudeamus", "virgo prudentissima ... quasi aurora", ...) I would decide for si bemolle, especially as there is a very prominent (turning note) F afterwards, and a si naturale would lead to an awkward tritone.

Note that there is even a third option that I have encountered frequently (?) in Renaissance cantus firmus settings, i.e. replacing the si (B) even with an ut (c). An example is Isaac's famous "Virgo prudentissima" where he does this at the phrase "quasi aurora". I doubt that he would have deliberately changed the cantus firmus; it is more likely that this actually was the planchant version sung at the court of Maximilian I.