Deb Lyons kindly got back to me, and it turns out that she is not the singer of the Fisher-Price Kick and Play Piano Gym (although I am not the first parent to email her and ask). She did record songs for two other Fisher-Price toys, but not that one. So the search continues!
She also satisfied my curiosity as to the payment structure: whereas you might be able to negotiate a bigger deal if you are the recognized voice of an established toy or character, most toy vocalists receive a one-time fee.
I have emailed Fisher-Price to ask if they will identify the singer. I’ll let you know what I hear.