I don't think Michele Lee had much clout in Hollywood, especially not back then, but neither do I think she originated the idea of having Rosemary sing "I Believe In You" as a solo ballad before having Robert Morse as Finch sing it with the chorus of execs in the washroom, the way it was done in the show.
"I Believe In You" had become the big hit of the score, and I'm guessing it was decided early on that whoever played Rosemary would get to sing the song as a ballad, so the movie wound up having two full performances of the hit tune in two different styles. And certainly, Lee may have signed on for the movie largely or only because she would get to sing "I Believe In You," especially since her character's other two big solo songs -- "Happy to Keep His Dinner Warm" and "Paris Original" -- were cut from the movie.
P.S. The show version of HOW TO SUCCEED does have Rosemary sing one verse of a ballad version of "I Believe in You" as a reprise, late in Act II, so there is that precedent for the way it's done in the movie.