Since I’m a recovering Protestant, may I chime in?
Two points to make here:
“The Believer’s Prayer” or “The Sinner’s Prayer” is not a pro-forma prayer as a Catholic would recognize (The Our Father, The Hail Mary, etc), but rather a broad brush term for the paryer invoked when one “gets saved”.

Losely based on what many evangelical’s would call The ABC’s of salvation: ask/admit, believe, and confess. Many would also point to “The Romans Road to Salvation”: four disconnected verses from Romans (I can provide specifics should you require them).
Where is it in the bible? It isn’t. Then again, where is the 2nd half of the Hail Mary? The Chaplet of The Devine Mercy?
I think you see my point.
-Michael