I have seen and been involved with different situations where the use of Llama/donkey(s)/LGD is the best combination.
Grt Pyr dogs are very hard headed, strong willed (like lots of breeds of dogs) and need very strong rule from the people on the farm/location from what I have seen. Twice the young GRT Pyr dog attacked me, and I blamed the owner, not the dog. I also soon did some work with the dog and it was about 50% better right away. The owners took way more work and after 2 sessions were only about 5% better! hhmmm…
That dog was only a puppy, although big, neglected and non-trained for anything. The owner read an article on-line that said this is the best guard dog for chickens and bought one! He and His wife did no research at all on big dogs, or on this dog, they raised little yappy dogs (poodle looking things), all of these dogs, barked all night, every-night, everyday, all day….. They also wondered why the neighbours did not like them very much.. hhmmmm again…
Donkeys with Llama(s) seems to be a good mix but you also have to have the Big Guard dogs as well.
All dogs need You to have training or instruction on how to best deal with them. Beating a dog does not make it a good guard dog (saw that from experience very sad), it just makes it Mean spirited and rightly so or takes the spirit right out of them. I have seen Grt Pyr dogs that are very non-threatening, but attentive to humans who come onto the property, they watched and were concerned with the humans, but so long as the person did and acted normal with no bad intentions they were fine.
Other large and mid sized breed dogs, work great as well. My personal experience is that any dog raised or trained properly will not hurt your livestock, period. The human is the worst factor here as they think they know everything and are reluctant to learn or change their ways. Large dogs take up to 3.5 yrs to become adults some mid sized breeds take from 2-3 yrs. Consider each stage of the year to be similar to ours 1st yr, kid, 2nd teenager, 3rd young adult. Get the dog, per your thoughts, lifestyle, interest and interaction levels.
Edit.. I am thinking that you might be able to find a couple of mature, older dogs, who will be good with the animals so that the little pups get trained by the adult dogs as well as from the humans, this might be good too. You might just help save some older dogs lives as well!
Grt Pyr and other large breed stock and guard dogs work great if understood and treated with respect and understanding, my personal favourites are as follow, Rottweilers, English (Bull) Mastiffs, Belgium Malinois / Melamois, then Great Pyranees (of which I saw lots in the mountains of France and Spain in the Pyranees Mountain ranges doing a great job).
I do not know about the 1 rule for the Llama, but there are lots of farms in the area across Canada with only 1 Llama so it might just work that way. These places had almost all of the animals in very large paddocks all together. A small pack of donkeys, 2 or more and guard dogs, 2 or more seem to be the best mix. When I was young I saw border collies work in a team, 2 guarded and herded the sheep and goats away from the two coyotes, then the other 4 herded, defended, nipped at the coyotes right off the property.
Later as an adult, I also watched from a distance, from start to finish, when the two coyotes first sprang into action from a ditch, that the goats attacked them, then ran, then turned and attacked defended, while the donkeys, charged across the field to their aid. These Donkeys brayed all the way there, so the dogs, came running and herding at the same time towards the coyotes as a team. We watched as the 2 donkeys put the coyotes in high gear in the opposite direction as 2 of the guard dogs, herded and stood guard with the sheep and horses. The Llama was sort of beside these dogs and watching the coyote as well. The other two dogs caught up with the donkeys and all of them chased the Coyotes off the property. I do not think those two came back for quite a while.
Now how did I see all of this I was driving by at the time, there was a curve in the road around this small homestead. So I watched this all as I was driving up to and around the small homestead. I also pulled the vehicle over very quickly, grabbed my ever present Compound Bow, with its quiver of arrows that were always mounted to it’s side. Why, because of the little kids in the yard. The two dogs who stayed back were also guarding, protecting the little kids who were in the yard and watching all of this as well. The dogs made sure the kids did not come into the pasture, by barking at them too! The Homesteader came out, saw me, I pointed to the field where the action was, and we watched together as this all occurred. He said afterwards that he used to get raided all the time by wild dogs and coyotes, but since he had put this combination of Llama/donkeys/dogs and perhaps the goats (can not remember what he said about them) in the paddocks that nothing had gotten to the chickens, turkeys, peacocks, sheep, and goats. We shared a very nice glass of homemade Lemonade together.
My way of thinking is that this is Permaculture team defence for your livestock. Like Permaculture in general, it is a phased array of many things working together in union and unison in a great big encompassing whole. Permaculture may take a while to get everything all together, but it works the best in the long run.