I'm not the OP but in our case: kind'a or as best we can. We live in dense, remote forest. The boundary fence I put in was to keep predators and varmints out, not necessarily to keep the dogs in. We've had turkeys, geese, ducks, chickens and piglets stolen by bobcats. We've have feed cans toppled and emptied by bear. Then Coyotes, badgers, ravens and on and on. So that was the reason for the fence and it solved 99% of the problems except for the ravens and wire/harness/hose chewing wood rats. The raven issue was mostly solved by putting chicken wire roofs over bird pens. We still may get a piglet stolen by a raven but its pretty rare although they still shop from tree tops. (Besides, I kind of admire ravens as smart and cunning as they are.)Have you done any boundary training with them? Fences are a human construct; they don't exist in an animal's world without help to tell them what you expect.
Here's a link to a great LGD training page: Farei Kennels Shepherd Training
So as far as the Catahoulas, as smart and stubborn as they are, the roaming/hunting issue was solved by only letting one out at a time and not on their own because the one dog is out to watch our back which it does. We don't need LGDs in a pasture to guard livestock as in the OP. But I sure understand the problem. And a Pyrenees is a whole different temperament than a Catahoula and hence the training is as well. A Catahoula in the woods is like an adolescent in a toy or candy shop - there's just too much temptation.
As for the OP, wouldn't it be cool if there was a perimeter wireless shock collar that would work with the electric fence? I've had good success with the systems for dogs when I lived in the city many years ago. With a smart dog (like a G.Shep) it didn't take very long at all.