Hi there,
I just want to make sure you are not confusing Stop and Pause. When the BeatBuddy is Paused, the screen is black.
The only way to pause is by using an external footswitch or a MIDI controller. The BeatBuddy does not play an intro if you pause and unpause using the footswitch. Unless I misunderstand you, the BeatBuddy already does this. Please make sure you are at least on version 3.8.0 or try out beta version 4.1.1
Let me know if this was helpful!