Any dog you purchase with the exception of one may be influenced by the GPs barking and then you would have two dogs that bark incessantly. The one dog I can recommend which has the least possibility of becoming part of the barking problem is a Besenji. They rarely bark if at all, they are a medium size dog, Smart, agile and willing to please.
However if your GP is barking due to boredom, that is easy enough to deal with by taking it for leash walks and correcting its behavior on top what you are already doing. Good Luck

Then you'd have two bored dogs :)

How about doggy daycare? He would have lots of friends to play with and be more tired in the evenings. Obedience training a few times a day will wear them out to. Mental stimulation is almost as good as exercise and it's good you're increasing the walks. At least an hour of walking a day for most dogs, more than an hour if he's got more energy than the average dog.

I would not have a big dog and a small dog, particularly if they are both young. Playing together could cause serious injury to the smaller dog, simply due to the size differential. I had a family member who owned a mastiff and got a whippet as a rescue; the mastiff ended up injuring the whippet's spine while they were wrestling in the back yard, the whippet ended up being put down because the injury was so severe.

That being said, I think most dogs do better with a companion. They are pack animals and do not typically enjoy being left alone.

I think doggie daycare is a great idea. Also, consider finding a neighbor with a same-size, same-age dog who can come over a few times a week for a play date with your pup.

Another idea would be a few weekly trips to a place where your dog can channel his inner lab and swim. Swimming is fantastic exercise for dogs, it really wears them out and satisfies the lab-instinct to swim.

A tired dog is a happy dog. And a quite one, at that!

Don't count on a another dog to stop your dog from barking. I had a Boston Mix for two years who rarely barked and got him a buddy in May. He now barks all the time on walks and in the yard when he sees people.

My son and his wife had a lab that barked alot!! They tried everything, training, exercise, etc. It didn't seem to matter if they were home or not; he just barked about everything!! They had a little mini dachshund too but it didn't seem to have any effect on the lab. Hope you can figure out a solution; it can be very stressful!!

FYI - My 87 pound Bull Mastiff mix had a best friend a Chihuahua mix who weights about 12 pounds. I was a bit worried about my dog hurting my friends dog, but they turned out to be the best of buddies, when my dog played he was very careful about the Chihuahua and neither of them got hurt. My dog was playing with another dog the same size and the other dog actually sustained a spiral fracture of her ankle - so I think some dog know when to lighten up rough housing - it all depends on the dog.

Try exercising your Pyr more. Doggie daycare is very mentally and physically stimulating and would be good.
Pyrenees are livestock guardian dogs and I don't think you are going to train that barking out or mitigate it with a buddy - he might just take the buddy as his "livestock" and bark even more while guarding him and may even guard him from you. Just tire him out so he's sleeping instead of barking.

I would worry about the safety of a smaller dog, if you got one. Pyrenees play very rough, they tumble and smack with their paws.

As for Basenjis, as a primitive (closer to wolves) breed I've read that they think for themselves and do what pleases THEM, not us (unless we have a treat haha).

Basenjis sound a lot like cats..lol

If the second dog is for companionship, I strongly suggest you get another dog approximately the same size as your barker. We got a second lab for a companion to our 2 yr. old lab. He is a male and at 11 mos., he is already somewhat larger than Sadie. We have to regularly tell him to 'back off' when he starts playing too rough because he outweighs her by about 10#.(He listens well and knows what we are talking about!)

They play constantly and for us, a companion dog for Sadie was a great idea -- takes lots of pressure off of us to constantly run and play with her and she sure does love her new 'toy'. Can't imagine it being much fun for a little dog to be bombarded playfully by a huge one.

I recently saw one of those "Dog 101" shows on tv and it had a segment on Basenjis. It is true they don't bark but they "yodel" which sounded almost like coyotes!! They seem to be a very independent little dog. Hope you can figure something out; I would try a bigger dog too if you decide to go that route.

OT but has anyone read the dogs in elk transcript? It's a funny story about a Basenji and I think New Guineau Singing Dog, 2 really primitive breeds, found an elk carcass, ate their wa inside it, and wouldn't get out. The poor owner had company coming and was posting on some message board on it, trying to get some advice. It has a surprise ending LOL

Here is a link that might be useful: Dogs in Elk story