First, the difficult part.
Don’t transfer to a public. At this point, you have no idea how much can get complicated from that. Transcript differences, course equivalency definitions, filled classes, a GC who doesn’t know you, your position of respect within the class or in a club/on a team, etc. And is this a Jesuit hs? They are generally better quality and straight shooters.
"I feel like this is definitely a big thing to put on an application that would probably raise my chances of getting into said universities at least a little bit." Not really. Lots of kids do time away, in one form or another. Or other things that have their own impressiveness.
Plus, over 4 years in hs, for a tippy top, you’re expected to have an increasingly strong trajectory, not just get in the right courses and maintain GPA. That includes the depth and breadth of your EC opportunities. You can’t cram all these elements into 3 years. Eg, I hate to think what trade-offs you’d make, if fitting in higher math or science means dropping something non-stem, but seen as equally vital to a tippy top.
You’d need to max out EC opps before going. That’s more than running something or winning some award. For stem, eg, it can mean an internship. Depending on what specific stem interests, it can mean some opps that are generally reserved for rising seniors, not juniors. Or, if premed, X years volunteering in a health setting, increasing responsibilities, becomes X-1.
You’re also trying to check the feasibility at a time when so much about applying is not well known to you. Successfully getting a TT admit relies on some strategy and a lot of awareness. Eg, not sure that, at this point, you know what the big essay is meant to convey. Many kids have their epiphany at a time when you plan to be done with your US hs years. You’ve “heard,” but can’t be sure, what club opps. Unless you’re already looking into specific Japanese programs/hs, you can’t know their stem offerings or the quality of those. (Or other bullets mentioned by @melvin123 .) Or just how difficult it might be to take courses entirely in a foreign language. (Do you know iy anything is taught in English?) Generally, what language one learns in a US high school comes nowhere near the tech terms, lingo or manner of discussing tech issues. And it sounds like you’d be relying on self study or some outside course to pick up enough? How green are you, at Japanese? Realize that this one issue can affect your grades there, too.
Generally, the kids I’ve known who did a senior experience, were 110% ready, in all ways. They knew the short list of foreign hs programs and the actual rigor, the parallel rigor to what they could have, back at home.
And, they were distinctly NOT set on an Ivy or MIT, but quite flexible about what targets. (Like flagships or a nice match college.) NOT looking at stem majors. They were looking at the opportunity, itself, that was their focus. Not looking for a tip, other than the relevance of studying in Japan to their future hopes and plans. How does this idea even connect to your future plans? (Tippy top adcoms will want to see how this makes sense.)
It sounds like your enthusiasm is so strong- and being fed here. You are right to be asking the feasibility questions. But please try to balance the excitement with some solid consideration of the challenges.
BUT. The opp is so great. I agree, you would need to time this just right.