he says she doesn’t leave his property without him knowing about it... I have a dog who stays in the backyard and never leaves it unless he’s on a leash.
I have to ask--how sure are you that your dog doesn't leave the property by himself, just like your neighbor's dog does?
For the neighbor's dog that "never" comes over--if it's a big deal maybe you could call Animal Control or catch the dog and take it to the dog pound. Of course, only if it's on your property at the time.