It is a safe bet that Saul (of Tarsus) was not in the Book of Life.
Well, yeah, he was. He tells us he was. He tells us God set him apart before he was born.
New International Version (©1984)
But when God, who set me apart from birth and called me by his grace, was pleased
New Living Translation (©2007)
But even before I was born, God chose me and called me by his marvelous grace. Then it pleased him
English Standard Version (©2001)
But when he who had set me apart before I was born, and who called me by his grace,
--Gal 1:15
But that is Paul speaking.
I guess the question is: why two names? Why Saul & then Paul? Which name is written in the Book of Life? Both Saul & Paul? Or just Paul?
Oh, I see what you're saying--I think. You're saying Paul was written in the BoL before he was born, not Saul.
He's talking here about 'God reveal[ing] his Son in me.'
Seems to me the name written in the BoL is Jesus. Saul or Paul is kinda irrelevant. It's the Son who's revealed that's important.

15But when he who had set me apart before I was born, and who called me by his grace, 16
was pleased to reveal his Son to me, in order that I might preach him among the Gentiles, I did not immediately consult with anyone; 17nor did I go up to Jerusalem to those who were apostles before me, but I went away into Arabia, and returned again to Damascus.
The Son was always there [that's why he was set apart before birth], but not revealed....
"reveal"G601
ἀποκαλύπτω
apokaluptōap-ok-al-oop'-to
From G575 and G2572;
to take off the cover, that is, disclose: - reveal.
This is the apocalypse--the book of Revelation.