You can listen to the DHCP traffic caused by the button press (the button is sleeping until pressed, so every press causes DHCP traffic) and use that to trigger any action you'd like. I wrote a simple utility[0] to do just that.
The longer term issue, since these are being discontinued, is getting your wifi credentials into the dash button. At the moment I use the amazon app to do that, but I'd guess they'll be removing that functionality. I recall reading someone reverse engineering that process, but I'm not sure what came of it.