I agree that 1) you shouldn’t have to build fort knox to overcome the neighbor’s inadequate control over their dogs AND 2) from a practical standpoint, making this The Big Issue with your neighbors isn’t worth it, when you have other solutions available.
Couple ideas: Coordinate with the neighbor where you get an underground fence that will be compatible with the fence he’s using, and bury a zap line all around your pasture. (I have no idea how pricey that stuff is, I’m hoping this idea isn’t ludicrously expensive). On your system, crank the power up as high as you want so the dogs will be given serious correction if they approach. Put the underground line fairly close to your fence, so they’d have to spend some time in the zap zone while trying to navigate how to get under/over/through the wire fence protecting the birds. In other words, don’t put in in wide open space that they can barge straight through and then be in the clear.
But no matter what, you’re going to have to get some cooperation from the neighbor.
Tell them that you’re investing in some pretty expensive enclosures for the chickens since their own fence won’t contain their dogs. But that if the dogs are going to be digging up your fenceline in a frenzy or running your horses into a fence, you need their help to figure out and pay for additional solutions. They could add a second zap line on their side.
Also need to discuss the plan on how to get the dogs off your property, if it happens. You can’t just chase them back home since they’re going to get zapped again when they try to cross back over the line. One idea is to ask the neighbor to pay for one of those 4x6’ chainlink kennels that you put on your side of the property line, so you can kennel the loose dogs and leave them there with a bowl of water until the neighbor can come get them.
It’s very likely that free range chickens just aren’t worth it, given these dogs and your limited options to restrain them. I know lots of people who came to that same conclusion–the cost/effort to keep them safe wasn’t worth it. I mean, I love our free range setup, but it’s not like chickens are going to actually eradicate the ticks anyway. They instinctively avoid wide-open areas to stay safe from aerial predators, so you’ll probably find they don’t range all that far from their coop. Unless you provide ground cover for them (shrubs, loose piles of branches), which now you’d have to mow/weedwhack around.