The oldest manuscript to contain Acts 13:9 is Papyrus 45, a manuscript dated to around 200 CE. Unfortunately, it's quite fragmentary, so the whole of verse 9 isn't extant.
However, the following words are: [κ]αι Παυλος, πλησθεις ΠΝΣ αγιου, και ατενισας ε[ις] / [k]ai paulos, plestheis PNS hagiou, kai atenisas e[is] / also Paul, filled with Spirit Set-Apart, and looked intently tow[ards].
The words "also Paul" translate the Greek και Παυλος/kai paulos which is the end of the phrase "And Saul, who is also Paul..."
So yes, the oldest manuscript of Acts 13:9 is evidence towards this phrase, which should be more than enough evidence.
The next earliest witness is Codex Sinaiticus (around 325-350CE), which has: σαυλος δε ο και παυλος πλησθεις ΠΝΣ αγιου ατενισας εις αυτον / Saulos de ho kai Paulos plestheis PNS hagiou atenisas eis auton / Saul but, who is also Paul, filled with the Spirit set-apart looked intently towards him.
As you can see, Sinaiticus and Papyrus 45 are almost identical word for word (Papyrus 45 has an additional και /kai meaning "and", so hardly anything important). And if Papyrus 45 had been preserved in its entirety, I would expect to have seen σαυλος δε ο/saulos de ho in the text.
Failing that, there's Papyrus 46, a manuscript dated to around 150 CE containing Romans, Hebrews, 1 & 2 Corinthians, Ephesians, Galatians, Philippians, Colossians, and 1 Thessalonians. And these preserve the usage of "Paul" in their text.
You'll find with many Messianics (from whom "The Scriptures" is a product), is that they have this bizarre hatred for all things Greek. And yet, the earliest and best manuscript evidence for the text of the NT, is all Greek.
So the reason that have Shaul instead of Paul is nothing to do with any evidence at all, only their anti-Greek polemic and agenda, which is a dreadful shame.
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