It works!

The project board, Trinket, and the slightly modified original circuitry all fit nicely inside the gutted body of the card reader.

The project box you see attached to the card reader with 4 wires contains a push button switch for input and a 9V battery that powers the Trinket. I made sure that the push button only triggers the song when the song isn't active using a boolean flag.

In addition to playing a song, the green LED light that formerly indicated a good card swipe now illuminates in time with the song. It is connected in parallel to the beeper.

Due to poor time management, I wasn't able to program a second song, but there is space on the Trinket to include at least one more tune.

Above is a small sampling of the song, which you might recognize. Appropriate to this project, the song goes:

I was just guessing
At numbers and figures
Pulling the puzzles apart

Lots of this project was guess and check.

I await a response from the judges of the contest.