The show doesn't revolve around Kim. In fact, the plotline involving Kim is secondary to the main action. The movie was a vehicle for Ann-Margret, so it focused very heavily on her. Having Allie Trimm sing the song at the beginning of the show would make it seem like the show follows her, when it actually follows Rosie and Albert.
A production I worked on back in 99 tried to incorporate the title song from the movie into the Entr'Acte starting with our youngest ensemble member singing it (she was either 11 or 12) and then being joined by the rest of the teen-girls. We even started staging it in rehearsals while waiting to get clearance to do this. Ultimately, even though our MD got enthusiastic response from Strouse and Adams, who really liked the idea, ultimately we had to nix it as either Tams-Witmark or whoever actually owned the rights to the song (maybe Columbia?) wouldn't grant permission.