I've always thought about doing stings from the other direction: with bait bikes. For a bait bike, you just need a bike and a GPS tracker with a battery life of a week or so (a poorly-locked bike will be stolen in a matter of days). The GPS tracker is inserted in the seat tube, and if you glue/oxidize the seat in the seat tube, the thief can't remove the seat to check even if they are smart enough to check for a tracker. You can even add security cameras where you set up the bait bike (outside your apartment or home is a great place) to get additional information. The GPS tracker then gives you a location of either where the thief lives or where they store their stolen property, which lets you bust them pretty easily.
Even in a bicycle-friendly town with 100,000 in population (ala Berkeley), I can't imagine that there are more than a few hundred bike thieves. It would take only a dozen individuals doing bait bikes to bring down bike theft drastically. Combine that with "Bike Batman" style vigilantism identifying stolen bikes on the used market, and I think it's possible to reduce bike theft to near zero in a local area.
Switching it on for a minute or so every hour if outside BLE range could help, but would make it harder to locate your bike fast. Maybe, it should start at a 100% duty cycle, and drop off over time? (Clock drift shouldn't be a problem. The device has GPS, so it has an accurate clock)
And you need to filter that 'motion detected' thing, or your bike will keep resetting itself, even before it can check the BLE connection. Or is that something that the sensor ICs can do for you, too, nowadays?