If the communication is bidirectional, you could include a GPS that only turns on when polled. That way, the GPS is only using power when you are actually looking for a lost dog.

It's not a tracker, it's an alert that your dog has left a perimeter at the moment it happens. It's a lot easier to track your dog down if you start the instant they get out of the yard.

LoRa can provide rough location data by tower triangulation, similar to our current cell phone location. Obviously not as accurate as GPS but better than nothing!

A sidewalk doesn't move so you can use it as a GPS. Since you can be way less precise it will take a lot less energy.