It sounds like you've thought this through pretty well and none of the obvious and simple approaches that I would have recommended sound likely to solve the problem. There are a few generic things to mention, although I suspect you've thought of most of them.
First, I'd reiterate that staying with a job that's seriously impacting your mental or physical health is a bad idea unless you either have no other option or only worse options.
Second, I think most people have a lot more options to make their current jobs better or at least less terrible than they think. I was an engineering manager for about 15 years, and I was always surprised at how little my employees asked for or negotiated. I tried very hard to make my employees' jobs as rewarding as I could, but I only had a few chips I could call in so I had to use them in the most critical situations, but I did have those chips. For example of what I mean by a "chip", I might have some cool R&D work to assign that most of my engineers would enjoy, but might only be able to offer it to one person each year. Of course if someone was going to leave, all of a sudden management would allow me to offer raises, time off, changes to assignments, or whatever we could think of. It cost a tremendous amount of time and money to get people trained, so we really wanted to keep people. If your school is having trouble staffing their teaching positions, they may be more willing to work with you than you think. I don't know what that would entail. You're in a better position to think of options (teaching 6 classes a day instead of 7?) that might help improve the situation than I would be, but I'd really think about what you could ask for that might improve your working life. That was not very well written, but hopefully the point came through the random stream of thoughts.
Third, if your current school won't do anything to help I'd consider other options to use the skills and connections you have to find a better job. For instance, maybe you're 2.5-3 years away in your current job and they won't offer any part-time options or work to improve your situation. In that case, are there any professional tutoring companies you could work for? Could you do private tutoring? Even if the pay is 1/2 what you get now, you might get more flexibility, fewer hours, and less stress. So it might stretch your FIRE date out to 5-6 years, but living a better life during those years might make it worth it.
Finally, I often see people post here that they have just option A and option B while ignoring the in between options. It sounds like you are facing down 2.5-3 years of work while you want to quit NOW. If you don't feel you have any other options, tell yourself you just need to get through this school year and the next, so a little under 1.5 years. At the end of that time, you might be able to quit if you add in some substitute teaching work, teaching just summer school for a couple of years, or doing some private tutoring either on your own or with a company. Or you might be able to scale up your 2nd job just a little bit. When you're within 1 year of FIRE, so many options open up. At that point just an extra $5k for a few years can get you through to the end. Or you might get to the end of the 1.5 years and decide that you can do another year. Or if you look for other options you may find that you could quit at the end of this school year and substitute teaching/tutoring/increasing the 2nd job just a little might be enough.
As I already mentioned, the biggest mistake I made before I FIREd was that I transitioned to part-time too late. If I had known how little it would impact my finances and how substantial the mental and physical health benefits were I would have done it much earlier. The 'stache was doing the heavy lifting at that point, marginal tax rates reduced the financial impact of going to part-time, and the extra day of "free" time was better than expected. I think the last couple of years should be the best working years of your life if you use the power of FU money to design those years to your benefit.
Good luck!