Can you find a lost Apple Watch if the battery is depleted?

Sure. Probably with a metal detector, if you’re close enough. Otherwise, look for it with your eyes, feel around in the couch cushions, whatever.

As for what you are hoping for, no. When setting out to find a device with a depleted battery, there’s no power for the GPS or Wi-Fi radio used to locate the device, and no power for the other radios (Wi-Fi or cellular) used to communicate remotely, and no power for the processors and memory that are used to coordinate all the related processing.

There’s no additional or “extra” or hidden battery that can power all that.

Which means looking for the Watch “the old fashioned way”.

Not using technology. You can try to remember the last place you no you had it and retrace your steps from there.