Shoot 'em from helicopters.
They are neither livestock nor wildlife. They're introduced pests. It may sound cruel, but good environmental management should focus on the health of the environment, not sentimental attachment. I would encourage private interests to take the horses if they want them, and I would prefer reducing the population with birth control, if a way can be found to implement it. But if all else fails, letting them tear up the range and warehousing them in pens is cruel to the animals or damaging to the environment.
Perhaps re-introducing the wolves or other predators that once preyed on big mammals in that area would also work, but I doubt the ranchers want to hear that either.