Leave to better job after contract sign

Teaching Eng for 8 years, most recently at Eikaiwa and thus I'm on Specialist/Humanties/etc visa. I signed a contract with a dispatch company to be a lead teacher, not ALT, at junior/senior high school in Tokyo. Job contract says term is 9/1-3/31 (6 months), but first day would actually be 9/3. However the process to change my visa to Instructor hasn't begun; when I asked the dispatch company that hired me was like "ah we'll get around to it", they might planning to apply for it the day before when we're scheduled to meet again.

Meanwhile, I got a dream job offer in a video game company that I've always wanted. It would require me to be on my current visa (specialist/humanities/etc). Therefore they are not competing jobs so nothing about competition should matter. But the contract with the first says stuff like: "Instructor is required to work the whole contractual term. Failure to do so and the instructor is liable for any inconvenience/loss the company suffers." and "If company or instructor choose to terminate the contract during the term of contract, 30 days advance notice should be given. If no notice or short notice is given, the company is to be compensated by the instructor for the loss incurred."

I might not be able to start the game job until a month later anyway, so I thought I'd just give a 30 day advance notice, but it seems unlikely I'd be able to change my visa to instructor then back to humanities in such a short period. Should I just say "here's my 30 day notice" and let them figure out that a visa change won't be happening on their own? Could they sue me, and/or how much in "damages/loss" could they legally demand in the worst case scenario?

I guess it is up to how strong you feel about the other job vs. good work ethics. I mean, things like this happen.

...after signing the contract, but before the actual work starts, right? I am assuming that you haven't received any orientation/training at the new place yet.

Considering that though youfve signed the contract to start work on Sept. 1 or 3 that they are not even beginning the resident status change process, you gcouldh say you donft have confidence in that contract (the employer-to-be doesnft sound serious about it), and could just give the notice gnowh/back out (before the contract term starts) to move on to the other job.
And since they havenft started the resident status change process, you have no need to switch your resident status, correct?

If they wanted to give you a hard time, they might try to get you to work for them even just for a month (atmosphere would not be friendly, as they know you are leaving); they could try to sue you for whatever damages they feel you owe them for the trouble to hire a new teacher. In your case, informing immigration that you left before you even started would be at their risk, as they never began the process for changing to the correct resident status. Best wishes.

Could they sue me, and/or how much in "damages/loss" could they legally demand in the worst case scenario?
You signed a legally binding contract. Yes, they could sue, and if I was them, given it is 30th August and you are supposed to start this week, definitely they should be taking steps to enforce their rights.
Damages/loss, probably they will suffer 100% of what they were going to pay you - expecting to make that much profit off your time would not be an unreasonable assumption - and for the full six month duration.
If you seriously want to back out of your contract you seriously need to get proper legal advice.

I would not accept a job and immediately give them 30 days notice. They actually need a teacher and that puts them in a horrible position. You can really jeopardize their school/business. If you have a job that won't start for a month, just sign the contract with that company and wait for a month for your job to start.

@JapanCustomTours: I don't think they could sue him considering they are asking him to work illegally by telling him he needs to start work on Sept 3 knowing that he needs a visa status change and that they have not given him the necessary documents to start the process, which will not be completed by Sept 3 even if he applies now. If he is concerned about this company, it would be justified. If they really want a teacher, they should be taking it more seriously. Of course I don't actually know, but most companies wouldn't sue if it meant exposing their own illegal activities...

@Rabbityama - without knowing the legalities of the business he is going to, I made the assumption that they were legitimate, and would pursue damages. I know there are companies that are happy to break the law, and if that is the case here, the OP obviously didn't do very good due diligence on them before doing a contract with them.