The reason employers for non-teaching jobs don't accept anything lower than a certain JLPT level should be obvious...those levels are not sufficient enough to do one's job. So, unless you get lucky enough to find an employer who is willing to dip lower in the JLPT scale, the vast majority of jobs here will be beyond your reach.
You need a degree related to the field. You have one. This will get you a work visa in that area after you get hired.
To get hired you have to prove to employers you can do the job better than a Japanese can. That means being able to communicate well enough in Japanese first, and add to that having the abilities/experience/skills they want. Programmers are a dime a dozen here, and Japan often goes to India to farm out its work (or hires Indians to come here), largely because they will work for less.
Putting on a career guidance counselor's hat, I'll tell you that unless you find something in your field within a year of entering the job market here, don't even think of taking a job that isn't related, even if it gets your foot in the door. Gaps in one's work history are serious, whether you want the job in Japan or back home.
So, go ahead and get a teaching job to tide you over for a year if you are diligent and persistent enough to job hunt for a programming job while you learn the ropes of teaching. It should seem like very hard work, but that's because it WILL be. You will be physically here, which means you'll be available for any interviewing, but you still need to prove to an employer that you can communicate well enough and that you have the background needed.
Communicating means speaking and listening, reading and writing. Office memos, phone calls, weekly meetings, reports/updates, etc. Programming engineers in Japan are notorious for avoiding the use of English, by the way, so you have a very long and hard road ahead of you if you think you can get by with a smidgeon of Pimsleur or Rosetta Stone and whatever work history you have now. Whether you own a zillion homes in Michigan makes no difference whatsoever. Would you hire someone and want to work beside them in the US if they couldn't understand daily WORK-RELATED conversations, meetings, reports, email, etc.?