I think the issue is that it didn't happen on the homeowner's property, so technically it's not a homeowner's insurance claim. Nor did it involve their vehicle, so it's not an auto insurance claim, either. That means your only recourse might be suing them directly. They should
be legally liable because they didn't secure their dog, but obviously you'd have to spend a little money on this. Depending on the amount of the damages, you could take it to small claims court, where it wouldn't cost you too much to get this taken care of.