They are called GPS trackers or GPS loggers. You can find some that save coordinates to a microSD card and optionally send the location via cellular connection for about 10 dollars on AliExpress.

Any cheap Garmin watch should do the trick

And if you want to spend more, Garmin even makes a range of dog tracking equipment: https://www.garmin.com/en-US/p/965617

You know honestly? I have thought of using a XOSS cycling computer (https://www.amazon.com/XOSS-Speedometer-Accessories-Waterpro...).

I have used them before on various bikes and they work just fine. Battery life is about 25 hours, it is weather resistant, and then you can sync them after you record an activity. And at less than $30, if it gets lost, it isn't the worst thing in the world.

You could definitely use meshtastic devices to do this

My dog has a Fi GPS/cellular collar. You could just get that and leave it on at night and it would show you where your dog wanders around.

This person specifically wants offline.

I have a TK913 GPS tracker for my ebike, which is configured to send location reports to my server running Traccar[1]. I'm very happy with the setup, it Just Works. I can check where my bike is at any point from the web UI from my phone. It supports A LOT[2] of these cellular-enabled GPS trackers.

[1] https://www.traccar.org/

[2] https://www.traccar.org/devices/