From my experience with Great Pyrenees (pure bred), they do wander, a lot! And they will tear down a fence to go wander and do what they want. We had more than 5 acres in one area fenced for just 2 dogs, chickens, and sheep. They didn't give a crap about the animals and cared more about exploring. Now, everyone has different experiences so this isn't ALL Great Pyrenees.
Currently, we have a 2 year old female Great Pyrenees/Akbash mix. She is super good with animals, but escaped in the past if someone left the gate open so she could explore the country. When we fixed this issue, no problem. She has been great ever since. Now, we also have a Great Pyrenees/Anatolian cross male that will be a year this year. He is also great with animals, and even when we didn't have a fence up while he was a puppy around the house, he stayed in the yard and never wandered. He has yet to test the fence and is happy and content to stay with the animals.
I will say I'm not sure about the mix you are going to look at. Not only would I worry about the Great Pyrenees mix in it, I would also really worry about the Black Mouth Cur mix. I am not sure how they'd be around chickens or small animals since they're used for hunting, not protecting animals.
My recommendation is either a pure bred Anatolian Shepherd, Anatolian Shepherd×Great Pyrenees cross, or a Great Pyrenees crossed with something used for guarding, like an Akbash.
But whatever you choose, make sure to introduce them to the animals at a young age and know it does take a little time for the big beauties to mature all the way.
***Edited to mention our dogs are not food aggressive, that also comes with time as well as training if it is shown in the puppy stage.