You're essentially asking if you can add a light in series with other lights. Almost all ac lighting is done in parallel, thus the live and neutral at each socket.
Technically it would probably work, but if your light can't handle the full load current of the other lights on the circuit, it'll burn out and then cause those lights to go out as well.
This.
Without further context on the install location it is hard to say but it seems like this should be something you could avoid doing with a slightly different design.
Of course… Something was nagging at me that I shouldn’t do this. Your reply’s mention of being serial was the source of the nag. I’ll explore damium’s current sensing idea instead. Thank you!
No, they wouldn't burn out, the garage lights would only work when the light near the panel is lit and both lights would be dimmer if lit at the same time.
I'm not an electrician so I don't know if it would be up to code (very likely is not) but you might be able to use a current sensing relay to trigger the lights. For safety I would add a local disconnect as well.
Great ideas! This sounds like a much better approach. Thanks!