That would be nice to have, but we’re not there yet. Different lithium chemistries are being tested for implantation, but I would not consider any of them safe enough at this point. Even the Lithium-Iodide cells used in pacemakers are questionable.
Charging via a port through the skin is courting infection. Wireless charging is an issue for the reasons already stated. Plus, charging batteries get very hot. It would be rather inhumane to subject your dog to that.
As for “other energy supply”, present energy harvesting methods don’t produce enough current to operate something as power hungry as a GPS antenna. Piezo, peltier, and betavoltaic elements are neat, but don’t do the job.
I am not fully aware of your situation, but it seems like the easiest path forward is training the animal.