Ok, I know this might sound bad, but maybe you might consider trying it.
I agree with your parents that the wood will not last as long as the chainlink. Plus, dogs have a way of getting though the bottom and loosening parts of it. But also consider how long your neighbor will be your neighbor. That said, I would do the whatever it would take to preserve my privacy, and my climbing vines, and my money. I wouldn't want to remove a chainlink fence because you might want to put it back up someday. Or you might want to sell your home, and the chainlink might be considered better than wood, which rots eventually.
If I had your problem, I'd put neutral grey heavy plastic tarps on your side of the fence with white lattice over it. You can use the lattice for your vines and for trellising. It will look hideous on his side (and from what you wrote, he won't care), but on your side it will look like white lattice against a neutral greyish background and your vines and trellised greenery will be over that. It will provide some shade for your foliage, it will block his poison, and it's not permanent or expensive. You won't have to worry about things going under the fence or through the slats like you would with a wooden fence. If you ever put up an arbor, it can go with the rest of the lattice.
Before you totally chuck the idea out the window as trash, consider one piece of lattice and one tarp, and see how it looks.
Oooor, you might think this idea totally sucks and I'm off my rocker. It's okay, it was just an idea
I just hate to think of the thousands of dollars your spending for a freaky neighbor.
BTW, the "tarp" I'm thinking of is the heavy covering they use to cover boats and trucks, not the flimsy blue tacky light ones. They're more sturdy and they look nicer.
