How to remove roundleaf greenbrier (Smilax rotundifolia) – a pernicious thorny vine

Question

Thorny vines keep popping up in my front lawn and apparently it's roundleaf greenbrier. I dug out the extensive rhizomes the best I could, but I discovered that it's coming from the property next to mine (which borders a church parking lot) and there are a few vines with roots right under the fence. I cannot dig those out. What is my best option to kill this thing so it doesn't keep coming back? I'm afraid to apply herbicide because it's only a few feet from my mulberry tree which I am quite proud of. Is there a more natural way to kill the roots at the edge of my property? Epsom Salt? Vinnegar? I would also prefer to not kill the surrounding grass.