Integration with Home Assistant

Looks good, although I guess you could keep advertising battery/temperature even when the button is pressed, and should probably advertise 0x0F/btnData.state all the time so home assistant knows the button is there all the time, just not pressed?

the advertised count reverting to 0 after the button is pressed doesn't seem fast enough

What exactly isn't it fast enough for?

It's actually a bit unfortunate that they chose to do buttons this way, because it feels like either you have a short timeout and you risk the button press getting never received, or you have a long one and potentially other devices think the button is pressed more than once. If the button had been advertised as a 'press count' that just incremented when the button was pressed things would have been much easier - no timeout would be needed.