Accepting your handshake is not necessarily a sign of good will, which was confirmed by the disgusting way he rejected your invitation for a compromise. You really don't want to be extending an olive branch to a neighbor whose opening line to a fellow neighbor is and aggression like that:
But now that you have the good intention to do so, at least don't do it under his terms.
This person had the nerve to come to your home and antagonize you. He thought he had won before even starting to negotiate like a sane person, you don't want to offer positive feedback on his insane attitudes.
Remaining calm was the best thing you could do, you can be proud of yourself. But other than that, some other answers are suggesting that you go for the submissive strategy, which is the same as treating him as your boss, your ruler... please don't let assholes rule your neighborhood, this is how they get to rule the world.
- Extend an olive branch under your own terms. That could be expert advice from you on how to soundproof his home. Don't help him with such endeavor as that could be taken as admitting guilt and providing compensation in advance.
- Don't go to him, let him come to your property.
- Prepare a hidden camera and record his actions, recording inside your property and public spaces is permitted.
- Ask your wife to be prepared to support you instead of accidentally supporting the stranger, by calling the police next time the man comes invading your property and yelling.
Not being used to this typo of situation you might think it's ridiculous, but the police knows this is exactly the type of situation where many violent crimes are committed, i.e. a person who lost their shit.
You will have the footage as proof. He will be discouraged to call the police and if he does, they already know who's the problematic one. (That is, unless you are in fact in violation of some civil code, I don't think you are but it wouldn't hurt to investigate your local legislation.)
What you could do in turn and in parallel, so you are not the as... problem. Is soundproofing your own garage. We all will silently thank you for it. * Sigh *