Three different angles:
The federal 4th amendment prohibition on unreasonable search & seizure of "persons, houses, papers, and effects". At first blush, the open fields doctrine pretty much sinks this argument for them under traditional evaluations (here is a good 1 page synopsis in a 1983 SCOTUS case applying that principle:
https://supreme.justia.com/cases/federal/us/466/170/). There is a wrinkle that could be interesting here - the 24h/7d nature of video cameras left on the property. In some recent cases, some of the justices have become concerned about modern technology shifting the balance of 4th amendment considerations -- for example, while tailing a criminal's car is clearly ok as it is done in public, using a GPS tracker on the car is a 4th amendment violation as is accessing a wireless carrier's data about your cell phone location over time. So this argument may not be entirely dead if appealed.
What ever parallel search limitations in the TN constitution - I have no idea about TN law so who knows. I do know that at least Maine rejected the open fields doctrine for its state search & seizure law.
Simple trespass - Absent either some form of land owner consent or a specific TN allowance for govt trespass, the state officials would logically be "guilty" of common law trespass - however any such personally liability for this is quashed by qualified immunity so they can not be personally convicted or fined. However, if there was damage done while on the property, depending on TN state claims laws (laws that reduce the scope of sovereign immunity to allow for some claims against the state) the state could be liable for money damages for actual property damage. Also, if the court granted an injunction against re-entry in the future, it is possible that qualified immunity could be stripped for future violations.
So, given the facts of the article, their only real shots are a TN rejection of open fields doctrine or a the SCOTUS feeling uncomfortable enough with setting up the 24/7 video camera surveillance that they modify the open fields doctrine accordingly.