Er not sure what everyone else is thinking, if you spray the shoots on your side with weedkiller it will kill or seriously damage the whole plant including presumably the much larger bit in his garden, this is one connected plant and a systemic killer like glyphosate will travel through to the roots of the whole thing.
Whilst it is a menace to you spraying it and risking killing the whole plant would be willful damage.
In a perfect world he should offer to install a root barrier of some sort. You could Dig down a trench along the boundary and drop in some paving slabs or a heavy duty polythene mat, pond liner or similar.
In reality if he has no interest in sorting it out you either cut back and install a barrier yourself or go down the civil route contact the council etc but really that will probably go nowhere and create a huge mess of bad feeling and unwanted consequences.
The other thing you could do is tell him that you plan to spray any shoots that come into your garden and get his permission to do so, then if it kills the whole thing he hasnt much of a comeback.
Totally understand what you are saying, we have tried to talk to him, we have not approached him gung ho and told him we will kill it, but he has flatly refused and said its our job to maintain our garden as that's what we supposed to do, even though it is his plant ruining the garden.
In one way I don't see why i should take time out of my busy family/friend schedule and take time to discuss with council, after all he cannot be bothered for a few mins to help tackle this, this also puts me out as I have to go down to inlaws (hour each way) dig trench further, sever the roots on our side, dig up all the roots growing through grass. not a job I really wanted to take on and waste time as I have other things going on that need to be dealt with.
However, I will mention to him like you said, I plan to spray on our side regardless of what he says. Just pee's me off when people like this cannot be bothered.