The BeatBuddy doesn’t have MIDI commands that activate depending on the state of the pedal.
If your MIDI controller can receive MIDI commands and change its internal state based on MIDI input (I doubt it), then you could do something there. But even then, MIDI doesn’t actually have a pause command. MIDI has either start or stop, so it would be difficult to support “when paused”.
If you drive most things from the MIDI controller, and the controller supports “pages”, then you could trigger a start from the controller and then switch to a page that has your “when playing“ options. Then trigger a pause and switch to a page that has your “when paused” options, and the same with stop.
If you have a 6 button MIDI controller, you could probably have 4 buttons available in each state.