That Tennessee fence laws text is very confusing, on one hand it says "Tennessee is considered a “fence in” state. This means that a neighbor must fence in his property in order to keep escaped livestock out", on the other hand it says exactly the opposite: "Livestock is not allowed to run-at-large in Tennessee. Owners who willfully allow this to happen could be charged with a class C misdemeanor. Any person who suffers damage from livestock running-at-large could have a lien placed on the animals until the damages are recovered. Fences enclosing livestock should be made of sufficient material according to the law."

So basically, it means there is no fencing law you can rely on, you need to consult a lawyer to check what are the legal precedents, for instance. In the mean time, document every "invasion" with photographs, contact the neighbor in writing for a resolution, call the sheriff every time in any case and file a class C misdemeanor and a lien on the cows every time if your lawyer okays it. After a few of these, let the court decide how the problem should be resolved definitely, like installing a fence paid between the neighbor, you and your landlord, maybe, or whatever they decide. I'd probably just fence my side on my own and be done with it, might cost less than a lawyer.

But yeah, if the cows are not fenced and are not fed enough that they feel the need to escape, that neighbor is probably committing a lot more infractions to livestock laws, might be time to let the law look into it... I'd also go to the sheriff and have a straight talk to him, asking if he is favoring his deputy and his deputy's father over the law. He would be committing a serious conflict of interest if he did, and if he is honest, he will talk to his deputy and let the deputy talk to his father, since talking to the father achieved nothing. Don't stop putting pressure on them.