Using fluctuating voltage as an input
I wonder if there is another pin on the board I could read


I guess that would depend on the particular doorbell. The transmitter must send some representation of a button press to the receiver - a pulse train for the duration of the press, or perhaps just CW: carrier on, carrier off - which the receiver somehow passes on to trigger the audio. Look at the receiver's output side to identify that trigger signal and pass it, or a processed version of it, to the Arduino.