Don't know if anyone else has figured out a good way around this yet, but it seems that the Voice PE triggers the button press event when it's powered back on and connects to HA.
This means that the automation for whatever the last button press event type was (double_press, long_press, triple_press) is triggering every time I unplug the device and plug it back in again.
My automations are set up in the way recommended by the Voice PE Docs:
entity_id:
- event.bedside_voice_button_press```
```condition: state
entity_id: event.bedside_voice_button_press
attribute: event_type
state: double_press
................```