It's not private property. Barnett owns 360 acres there and has a grazing lease on 22,000 acres of federal and state land. These events were on the the leased land.
This is not the first suit that Barnett has lost and had to pay damages. The first one was by a Mexican American family who was hunting on the leased land who were held at gunpoint. In that trial there were witnesses testifying that they also had been held at gunpoint by Barnett.
It was only after Barnett lost the first suit that this suit was filed.
But as to the 'leased property'. If he's paying the G for the leased land, then only the owner, 'Lessor' (Gubmint) has the Right To Enter without the consent of the Leese.
It's just as if you're leasing an Apartment. You still have Private Property Rights, or a Right to Privacy under the lease. And one Right would be No Unauthorized Entry (Trespassing).
This guy may be a Flake, or maybe not. But if he's paying to use this land he's responsible for it. Like cleanup after these 'immigrants' who trash it.