You're in a difficult situation.
Best I can say is become part of the town in a dozen different ways.
Local fire and rescue (volunteer!)
Local library (volunteer!)
Kids in local school? (volunteer!)
Join a church (volunteer!)
Kids in local scouting groups? (volunteer!)
Attend local municipal meetings (introduce yourself and always be polite!)

When I say volunteer, I mean do the scut work that always needs to be done, not telling everyone how you did it better in the big city. If the community as a whole thinks positively of you and yours, it's easier. You'll earn social credit for when you need it.

Unfortunately, it will take ages.
Equally unfortunate, you're correct. More people will be burning plastic or other obnoxious things because they need to in order to keep warm in the winter.

How old is he? You may have to wait it out. And plant more yews as screening.

Best wishes to you and yours.

He might really enjoy burning things, so less likely to take you up on the offer, but maybe you could offer to take _all_ his plastic for recycling & disposal.

I can understand him feeling landlocked when he previously had an unoccupied side for a feeling of privacy and a psychological way out. (I live in a densely built part of the city and feel better when the houses on either side have been empty.) Maybe he has been living there since his was the only house around and the newer houses have really changed things?

I imagine he put up that fence for privacy rather than to contain his sound. He might like hearing that you intend to let vegetation grow up on that side of your property, especially if it was wild before your house was built.

Are there others who live with him? Just keep mind that his housemates might not bee too fond of plastic burning, either. Perhaps any offer to take his plastic could be made through a housemate. Ex: "Hey, Sandy, could you tell Bob I could take your household plastic waste off your/his hands for recycling and disposal? It'd be no trouble for me."...Sandy thinks"Hoo! I would be so glad if I could get Bob to stop burning that nasty stuff; it chokes me." Sandy says, "Jbucks, that's a really nice offer. Thank you. I've been telling Bob not to do that. I'll talk to him about your offer."

You said you had been putting out feelers with the other neighbors. Talk to the ones who live on the other side of your neighbor. They've no doubt had to deal with the smoke in the past. Ask them how they resolved it.

Also try to see if there is one who is on better terms with him than you are. You can then mention to that person that someone in your family has health problems from the smoke. (Which isn't a lie, plastic smoke is toxic and breathing it on a regular basis is going to give you problems, not to mention start making your house smell.) Tell the friendlier neighbor that you tried to speak with the annoying neighbor but have had no luck, so you are thinking of contacting the city about it. Perhaps the message will get up the chain to him, and he'll consider that modifying his behavior voluntarily would be better than having to deal with the city.

My guess is that he's had issues with this with the other neighbors, that's why he only burns it when the wind is in your direction. It also sounds like the fire and burning is his way of dealing with stress, either from family issues or yes, even PTSD. If that's the case then taking his way of dealing with stress away is going to add even more stress. You don't mention if he seems to be having family problems as well. The burning may just be the tip of the problems.

I would also start going to your city council meetings immediately. You should do that anyway when you move into a new neighborhood but in this case its even more important. Go sit and just listen to what issues the city is handling. Meet your city representative too. Then the following month, if you haven't had any luck getting him to stop, go to the next city meeting and discuss it with your city rep. Be sure to take some cell phone videos of him burning the plastic and especially of the smoke though, and start documenting the number of times he does it in a journal, so its not just a he said/he said situation. Him doing it once a month isn't as much of an issue as him doing it 3-4 times a week.

Ultimately, letting a bad situation fester is only going to make it worse. While I understand you don't want to escalate things, being new is actually an advantage for you. You can set boundaries now easier than if you had let it slide for many years. Having your next door neighbor pissed at you is bad but you being pissed off at the neighbor and not doing anything about it, is going to blow over onto your relationship with your new home, and with your family.

ADDED: You don't mention, is this suburban or rural?

Sometimes we all need to vent, I encourage anyone who visits here to feel free to do just that.

And contrary neighbors are going to be a continuing fact of life in the Long Descent. Learning how to (more or less) peacefully co-exist with those who live around you is a skill we've unfortunately let slide, much like the rest of our tools for living in communities. Our techno civilization and consumer society actively promotes people becoming hermits as a way to control us and to get us to spend money.

A side affect of that abandonment of curtesy, or at least that tolerance, has be that more and more people feel that their wants are all that matter, and that taking into consideration how your wants affect others, doesn't matter.

One of the reasons I put the Twelfth and Thirteen Circles in the line up, was that relearning critical thinking and community building are going to be more important than gardening and retro skills in the collapsing world we're headed for. The Green Wizard who is skilled in subtle persuasion and in how to lead groups to making decisions that benefit the most people, not the few will be the ones who help their community survive as a whole.

We'll revisit this subject as time goes by.