I don't have a dog in this fight,

, but here is my :2cents: .
Dogs are instinctively protective of their territory and will protect it. They will make noise and posture to drive the invader off and will often pursue it a short distance to ensure it doesnt return. This is a trait of a good watch/guard dog, and training will limit the pursuit to specific boundries.
As the owner of a watch dog, I am endeared to those traits. I am also protective of my dogs and do my best to keep them to my property by setting their boundaries inside my property lines using fence and for one dog and boundary collar.
As an infrequent neighbor, the dogs are incapable of knowing you are not an invader to their territory and are reacting appropriately as far as they are concerned. Your best approach would be to become a known quantity to them and they will be less and less inclined to harass you as you become non-threatening to them.
I would approach your neighbors and endeavor to make friends with the animals. They will recognize your scent and accept you as belonging. They will be able to associate the scent of your family as belonging to you and accept them as well. After a while, they will ignore you as they would your neighbor and bark only enough to remind you that they are there.
Until that time, I would suggest "assisting" your neighbor to reinforce "your" side of the fenceline. From your picture it appears there is a 1320 foot common border between you and the dogs and a 660 foot line running from that common border toward your picnic area. It would seem to me that the dogs are charging that common border and you are afraid they will follow around the ends toward you. Unless they perceive an ongoing threat it would be very unlikely for them to follow that far. Most dogs will feel they have driven off a threat in fewer than a hundred feet, so a 1320 foot boundry would be more than sufficient, and if you chose to fence the 660 foot line, it would mean an additional chase of an unusual distance to get to you. If the dogs are not already motivated to jump or crawl under the fence and chase you a hundred feet to get to you, it is beyond imagination that they will run half a mile along a fence to catch you.
If you are willing to put up that much fence to protect yourself and your family, I think you would be safe enough without any further effort. If you do that and make friends with the dogs, I think everyone will be able to live in full peace and harmony.
As the dog owner, I would be hesitant to put such a hole in my dog protection by letting you become friendly, afterall they are there to keep other people off the property. At the same time it would make my life better if they quit barking at you all the time, so maybe it is a good compromise.