GPS drains a ton of battery, too much for this purpose. And even if you would make it so that you need to replace GPS batteries once a month, that wouldn't be enough. In addition to using the GPS signal to geolocate, the bass would have to transmit this information to you in some way. Using phone signal to text the location to you once a day or every other day introduces additional battery strain (the majority of power is spent on looking up a nearby phone signal tower). Geolocating and texting the location as rarely as once a week might save a ton of battery, but GPS hardly works indoors and your best bet is to geolocate while the bass is on the move. But to know that you'd need an accelerometer, probably active 24h unless the bass is being played (plugged in).
Anyway, as far as I know as an aspiring Computer Science PhD, not really possible at this point.
The ID chips you talk about are passive; they do not transmit a signal, instead you need a active reader device in near vicinity to make use of it. Unique, passive ID tags could be built into the body of the bass, between laminates or something. They're certainly small enough. But they'd be just as useful as serial numbers (that cannot be changed) and that does not change much. The buyer of the stolen instrument would have to actively scan the chip to notice that the bass was stolen, the same as with the plain old serial number.
The Tile thing looks interesting. It communicates through Bluetooth 4.0 which can communicate only within small vicinity, but if the Tile became popular that would be mitigated in urban areas through the app. Though, I'm unsure about how Bluetooth affect pickups. The Tile as it is could not be turned off while the bass is plugged in, that'd fix that issue.