The Tesla App will provide a push notification for when the battery goes into backup.
If you allowed push notifications for the Tesla App when you installed it then these notifications will already be turned on. If you did not allow push notifications then you will need to change your settings so you can receive them.
Unfortunately this would require very frequent monitoring of the app, which wouldn't be feasible in my situation.
When my power is turned off and then back on a digital clock flashes until it is reset. If the transfer to battery (at night - no solar) took a second or two, this digital clock would be a good alert.
If the transfer to battery was seamless, it wouldn't be an alert at all.
The push notification won't require constant checking of the app as it will appear as a normal notification on your phone, such as the ones you get for a text message or email. The switch over from regular usage to backup on the Powerwall is designed to be seamless meaning that most appliances would not even notice the changeover and things like digital clocks would not reset.
I am a bit of a dinosaur and don't have or need a smart phone.
I am starting to think that a small led light wired direct to the grid might be the best way to go. If it turned off, I would then know that the grid has failed and battery only would be in use. This would then provide the opportunity to turn off power guzzlers to ensure the battery would provide essential power for longer.
I can't think of any other solution.