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Please do not try to take down something that large with a rimfire round, which is more appropriately suited for dispatching small animals. Granted, you can kill larger animals, but you realistically face the prospect of multiple shots and/or close range shots or work with a sharp/blunt object to finish the job - a gory and inhumane death. Most likely, a shot with a .22 would lead to the dog leaving your property, wounded (opening you up to personal liability, I would bet, despite your documentation of earlier trespasses by the animal) and would probably not sit well with your family's conscience (nothing about your posts strike me as malicious). If you live in town, a real firearm may not even be a legal option - please keep your family safe by researching local laws regarding discharging firearms and what would be appropriate firepower for this problem.
The dog's owners are the real culprit here and unfortunately the dog is going to pay the price for that.
Edit: P.S. Glad Rocky is doing well!
Disclaimer: I am not a lawyer and my personal opinion is worth what it costs.