I really want to leave
For the purpose of this answer I'll take it that your goal is to leave, and the question is how. As others have said, though, you might have other ways to deal with ill-health than quitting your job. If you can, speak to a doctor. Stress can be treated/managed short of giving up work entirely (in fact for manageable stress I believe often the advice is not to stop work entirely, because work itself provides structure and goals). You ideally don't want to get to the point where you need to be signed off work entirely, but if you do get to that point then you want and need it to happen as soon as it's recognised. Avoiding that point might take some concessions from your employer (limited working time etc) as well as help from your doctor (advice or treatment).
I fear I might freeze up eventually and be unable to work on the tasks even if faced with fear and urgency (I guess that's the definition of burnout in a way)
Suppose that happens. It's extremely bad for you of course. It's not really any worse for your employer than if you won the lottery. So just bear in mind it's your own problems you need to solve here, not anyone else's.
I don't want to punish fellow workers who have done nothing wrong
That's how they get you. Employees feel empathy and loyalty to one another. Perhaps even without realising, employers can weaponise this as a means of manipulation.
Your responsibility to your colleagues begins when you join the company, and pretty much ends when you leave it (of course after you leave they are still people you know, but they are no longer colleagues and you no longer have any responsibility to them to help them with their job).
You cannot "punish" someone by leaving, because leaving is something you're free to do at any time for your own reasons. Even if it happens to inconvenience them it's not a punishment. Even if you did it because of something they did (which doesn't seem to be the case), it's still not a punishment because you're not doing anything to them.
I don't want to burn bridges with an employer who is very well-connected in my medium-sized city
If this is a "one-factory town", where they are the only employer (or the only one in your industry), then basically leaving the company means you're going to have to leave the town to find work. Is that doable for you? If it's not doable for you, then you should recognise this as a very strong piece of leverage which your employer can, and is, using to manipulate you.
Switching jobs can be scary, but your former employer almost certainly has a lot less influence over your next employer than you imagine. Usually, when you leave a job, that's it. Its effect on your life ends immediately and permanently.
You also might have observed at work that management grumble about previous leavers having let them down. If you have not observed this, then congratulations! There's little chance they'll do it about you when you leave. If you have observed it, think also about whether this grumbling has ever actually affected the person who left in any material way.
It is worth thinking, before you leave, whether you are going to want use any of your colleagues as an individual reference (i.e. not just the corporate, "we confirm X worked here from Y to Z"). If so then maintain that relationship. Don't just quit and run from them, then hit them up for a reference out of the blue later.
I very sincerely doubt, based on reading the room, that "burn out" will be accepted as a non-bridge-burning reason for quitting
Keep in mind that you don't need your employer's permission to leave, and you don't even have to explain why you're leaving. Of course you know this, but you may not be keeping it high enough in mind!
You just need to pick a narrative, whether it's "I have another opportunity" (technically true: the opportunity to watch daytime TV), or "I need a break", or "I have some options but I can't do anything about them while I'm working full-time". Or if you really feel uncomfortable with that kind of narrative, maybe it's because it's not really what you want to do yourself. Do you have a realistic prospect of finding some project or work to go into, perhaps temporary and/or as a freelancer, that means you can avoid burnout in this job without being unemployed? Doesn't have to be your long-term plan. It would mean you can give the standard answer to the question "why are you leaving": "I've found another job".
Bridge-burning in any case is not all-or-nothing. Just because someone thinks you shouldn't have left, doesn't necessarily mean they'll never get over it, or that you should expect any particular problems with them if you encounter them again professionally in future. They will have an overall impression of you as a former colleague, and your time and manner of leaving is one part of that. If they do hold grudges to a pathological degree, then of course that's bad, but probably the best way to deal with it is not to just continue working for them until they retire and no longer care what you do...
if I were to give my two weeks they would probably be unable to find and train someone fast enough
That's a choice your employers made when they decided what terms to offer you employment. If they felt they needed more than two weeks to replace you, then they could have written you a contract in which you were entitled to more than two weeks notice and (although the term many not actually be enforceable in law), they encouraged you to give them more than two weeks notice in return. They chose not to do this. They prefer the consequences of that to the consequences of a general agreement that more notice is preferable. You can quite reasonably conduct business with them on the terms they chose: so give them two weeks.
That said, you have the option to offer them a longer notice period if you want to. You can do what you like. So, if you go to them today and say, "I will be leaving at the end of December", they have a choice. They can require that you leave sooner, on grounds that anyone on the way out is dead to them. In which case it's them exacerbating any problems there might be for your colleagues. Or they can gratefully accept that they have three months to find your replacement. They even have the opportunity to call on you to help train your replacement. If you decide this is what you want to do, just make sure you're mentally prepared that they could give either answer. And, for that matter, be prepared for them to walk you out the door direct from their office, do not return to your desk. As long as you're quitting when you're ready to go, then you'll be able to accept any answer they give.
I don't suggest this because I think you have any obligation to give a long notice. You don't, but it might help you leave in a way that's softer for you in terms of making new plans, and/or dealing with your feelings that you're abandoning them. It's also a lot easier to decline unreasonable overtime once you have your exit date written down, so it might directly help with the burnout.