You couldn't integrate it into theirs, so you'd be looking at having to fence in the entire space you want the dogs in because the gap you'd leave would have to be big enough for a dog.

Which is hardly the end of the world, but it can definitely grind the gears to be spending thousands when you could fix the problem for several hundreds, if not perhaps several tens, of dollars if they wouldn't be assholes about it. But I would absolutely advise anyone in that situation to spend the thousands and not bank on the state not being assholes about it.

I can also add that you can't even be sure that if you got away with it for years that you will continue to get away with it. The particular issue that my father got into a bit of hassle with the state about is that he accidentally put a pole barn across the state line. Didn't mean to, but he misread the survey markers. Fortunately, they let him remediate it at his expense without pursuing the matter any further legally... but it means they did, after several years of it being there, come through a with a survey team and verify that nobody on the border was crossing into their land, and anything they found they made the residents fix no matter how many years the deviation had been there. All the laws about such thing between two citizens mean nothing to the state.