Hey there,
As of version 4.1.3 you should be able to use an external footswitch (like the Footswitch+) to change parts to a new song part from a paused state
Go to the settings, Footswitch > Footswitch 1/2 > While Paused and select the next part option.
Please let us know if this is a workable solution for you
Otherwise, MIDI can be used to start specific parts on the BeatBuddy on the fly, yes