You could create command(s) for push-to-talk in addition to, or instead of, the global "Joystick Button Recognition" option (or the "Override Joystick Recognition Button" profile-level option).
E.G.
Start VoiceAttack listening
Start Loop While : Joystick 1 Button 1 Is Pressed
End Loop
Stop VoiceAttack listening
Where the command is triggered by the relevant joystick button, and the loop is set to run while that same button is pressed down.