First I would trim the climbers in your
fence at the base, you can pull them out later. Then go talk to your neighbor, explain that the plants are coming onto your property and filling the fence, (show him the dead stuff you had to trim) offer to cut a small path before the fence and maybe plant a few non-invasive plants to take up the space and prevent the problem in the future, explain how mints and other herbs are tasty (ot whatever is appropriate).
If he says no, dig a trench that drains off your property and nuke it (6 inches on your side where you were aiming and well.. overspray of 18 inches on the other side. There is only so much you can do and if you plan to move then you have a responsibility to the next homeowner for a clean space. You can trench on your side and use a rod that perforates the soil on the other side. You offered an organic, safe solution.
Be fair and reasonable, offer some bounty, and if he just doesn't care, fix your
land first.
I really don't think we can train or teach ALL of the world, I don't want to hurt others (even if the won't notice) but it's my last choice.