I do not think you should send it. You have jumped when she asked and have done some pursuing. All of which is paying off for you somewhat, it seems. However, if she was temp checking and you send her the song it she will know she has you. I would point out Sandi's rule about not pointing out what was good in the R. Sending the song is kind of the same thing.
Me 41 W 33 M 2013 Suspect A 11/15 Confirm A 1/16 She moved out 2/14/16 Stepson 13 Stepson 16