It's important to remember that we are talking about differences that boil down to a single yod or vav, acting as a pronomial suffix:
אמונתי (my faith)
אמונתו (his faith)
There is no difference between "faith" and "faithfulness" in Hebrew, the same word means all these concepts as well as "trust", "reliance", so it is up to the translator to select which word to use in the target language, and this will be influenced by the translator's theology and interpretive tradition.
Now we know that the LXX was translated from Hebrew, and the question is whether those copies had a yod, vav, or no ending. And we also know that our current copies of the MT have a vav, so the question is whether our current copy of the MT is wrong, or whether the (earlier) copies of Hebrew that the LXX were translated from were wrong, or perhaps they are both wrong.
Ultimately the question is unanswerable, but we can at least cite some evidence and also dispell some myths about these textual issues.
The first myth is the notion that the MT is "the" Hebrew and that any translation must be a translation of the MT, rather than a translation of some pre-MT Hebrew text.
Such a view is no longer tenable after Dead Sea Scrolls were found with Hebrew copies in agreement with the Old Greek translations rather than the MT in many (but not all) cases.
The discovery of different sectarian versions of the same Biblical books in Palestine should forever put to rest the idea that the MT is "the" Hebrew or that there is a single Hebrew version of any book of the Old Testament, just as it would be ludicrous to think that there is a single Greek copy of any book in the New Testament. There are different textual traditions in both Greek and Hebrew, with scribal errors and corruptions in all textual traditions and all languages.
The second myth is that when a scripture is cited in the NT that disagrees with the MT but agrees with the LXX, that author "must have used the Greek rather than Hebrew". Indeed, there is no evidence that the MT even existed when the NT was written but there were Hebrew copies circulating in Palestine. In particular, Paul was a pharisee who spent much of his life studying the Hebrew scriptures, he was just studying much earlier versions of these scriptures than what is handed down to us in the Leningrad codex, and it is much more likely that his copies did not have the vav at the end than that he was intentionally ignoring what his Hebrew copies said and decided to quote a bad translation instead.
First evidence: Missing Vavs and Yods in Habbakuk
Onto Habbakuk, there is strong evidence that the MT introduced corruptions not present in those copies that the LXX and other witnesses attest to. See, e.g. Tracing the Pre-Masoretic Text of the Book of Habbakuk, especially the discussion of verse 2.5, where it is argued that the MT's היין ("wine") is an error and should be הון ("wealth, power, fortune") which also had the altnerate spelling הין, or in Habbakuk 1.4 where ַבגּוִֹים is a scribal error in the MT and should be replaced with בוגדים (an extra vav) to bring it in line with the LXX as well as other translation witnesses. Thus in two instances the MT added or dropped the notorious vavs and yods.
In fact, dealing with the vavs and yods during the repeated copying was so error prone that various processes of checksums were later introduced in order to try to avoid the chronic issue of dropped/added vavs and yods. However we know that these types of checksums were not being used at the time of Christ since they are not present in the Dead Sea scrolls. Thus these safeguards were only added much later to the MT, long after there was much opportunity for vav/yod mutations in the repeated process of copying for all Hebrew textual streams.
Second evidence: Other witnesses
Let's look at available other witnesses:
the phrase is not present (only fragments of two words were (yasherah nefesh) found)
- Vulgate (late 4th century translation from Hebrew)
justus autem in fide sua vivet.
but the just will life by *his faith*
But the righteous man shall live by *faith*.
- Targum of Minor prophets[AT] (Aramaic translation)
But the righteous shall live by the *truth*
So it looks like there are two textual streams, the MT and Vulgate which are based on Hebrew copies that have the vav, and the LXX and Syriac Peshitta, which do not. The targum is an intepretative translation rather than a literal one, and does more to elucidate rabbinical thinking about a passage in this case -- clearly the rabbis wanted to steer the reader away from any notions of "faith" entirely.
Obviously it's up to the reader to decide, but in my opinion older texts should be trusted more than later texts, so without stronger evidence to the contrary, I would assume the Hebrew copies that the LXX bear witness to are more accurate than the (much later) Hebrew copy that the Leningrad Codex bears witness to, and would thus say the balance of evidence points to the vav being added in the MT. However, this has no effect on theology or meaning. What would have an effect is an interpreters choice to use faithfulness rather than faith, which is no longer a textual question, as the same Hebrew word is used for both. Rather this is an interpretive question, and it depends on whether you believe the key issue is to wait, as per verse 3:
For there is yet a vision for the appointed time; it will give witness
to the end, and it will not lie. If it tarries, wait for it, for it
will surely come and not delay. (LEB)
Thus if verse 4 is a reference to those who choose to wait versus those who do not, a better case is made for faithfulness, in the sense of a bride being faithful by waiting for her groom. But if the meaning of verse 4 is less dependent on verse 3, but is a general principle, then "faith" is a better translation. In both cases, the meaning is faith, in the sense that the faithful bride who waits does so because of her faith that the Promise will arrive.
[AT]: Cathcart, K. J., McNamara, M., & Maher, M. (1990). Editors’ Foreword. In K. J. Cathcart, M. Maher, & M. McNamara (Eds.), K. J. Cathcart & R. P. Gordon (Trans.), The Aramaic Bible: The Targum of the Minor Prophets (Vol. 14, Hab 2:4). Collegeville, MN: The Liturgical Press.