When I built my genset (egad, yet another project to post details of) and built the controller for it, I made a couple of design considerations in this regard, Chris.
1. When the inverter (or any other 'start' source) requests the genset to start, the first thing that happens is, it turns on a loud beeper for 2 clear seconds before it does anything else. A kind of "get clear" warning.
2. When it's in bypass mode, it gives a short, sharp "cheep" every 5 mins to remind you it's offline. Sounds a lot like a battery smoke detector with a low battery, only louder.
The beeper is also replicated in the house. We know one long beep means the genset is starting. Five short beeps means it's shutting down. Continuous beeping 1 second on, one second off, means the genset needs attention - either it's failed to start, or shut down unexpectedly, or in danger of shutting down (high temperature, over temperature, low oil pressure etc)
It's not foolproof, but it's close.