I have a little bluetooth FM transmitter that uses a single button that you press and hold to turn the device on, press and hold to turn the device off, and press once to change the FM station it broadcasts to. Is there any simple way to alter a circuit like this to be "always on" and essentially short circuit the switch?
If it were some sort of toggle push button switch this would be simple, but seeing as it's a simple momentary push button used for other things, I have no idea where to even begin.
Try using a normally-closed pushbutton switch. When you apply power the unit will power up according to what you've indicated in the comments of your orignal question. The other actions invoked by the pushbutton might be edge-triggered and still work as intended.
Alternatively, connect a SPST toggle switch in parallel with the original N.O. push-button. Before you apply power to the device make sure the toggle switch is closed. Once powered up, open the toggle and use the PB switch as usual. Throw the toggle switch closed again to turn off the unit and leave it closed as you remove the power source. The toggle switch will then be ready for the next power application.