The name Carole Bishop was already taken in the actor's union, thus she changed it to Kelly. Several actors have changed their name just for that purpose.
A Chorus Line revival played its final Broadway performance on August 17, 2008. The tour played its final performance on August 21, 2011. A new non-equity tour started in October 2012 played its final performance on March 23, 2013. Another non-equity tour launched on January 20, 2018. The tour ended its US run in Kansas City and then toured throughout Japan August & September 2018.
You can. Thankfully, this was one of (if not the first) Broadway shows to be archived by the Lincoln Center Archives. Before the show transferred to Broadway's Shubert Theatre, it was filmed during its pre-Broadway run at the Newman Theatre... alas on primitive black & white video (hence the quality).
Here is Kelly as Sheila in the original cast of A CHORUS LINE:
"Does anyone know why she changed her name (from Carole) mid-career?"
She most likely had to change it when she joined either SAG or AFTRA. She has a string of Broadway credits as Carole, making it clear that her name was available when she first joined Equity. However, she only has one credit on IMDB as Carole, the rest are as Kelly.
I just appreciate that she took a whole other name, instead being at the forefront of the three-name epidemic that plagues us now. I refuse to believe and there's both a June Raphael and a Diane Raphael in SAG. Pick one name! Names like Mary Ann and similar excepted.