Yes, by replacing the battery, it would reactivate this AirTag.
It's possible that the owner did not enable Lost Mode on this tag. If your iPhone has an active Internet connection, it should relay the tag's information to the Apple servers. From their the owner would see it with the Find My app on their iPhone.
Chances are the dog's owner is nearby. Did the dog also have a collar with an ID tag? If not, it could also be possible that the owner had the dog chipped. I would suggest if the owner does not appear within a reasonable amount of time, that you can ask neighbors if they know this dog's owner. Should that fail, the next step would be to take the dog to a veterinarian to check for the chip.