I know it's a weird hangup to have, but stuff like the "Of course it's impossible for me to attend a regular high school" keeps coming up and it always takes me right out of it - like, why? Why would it be? Maybe it's weird to readily accept time travel but consider this weird bureaucratic situation to be strange, but seriously, what's keeping her from resuming her "normal" life path? She'd be surrounded by a completely different class, but that's no different from what a transfer student would go through.
Because any Japanese high school she'd look to enter would look at her paperwork, see that she's legally 21 and then politely decline. You can't just go back to a regular old high school if you're an adult in Japan (and probably many other places in the world) and despite the time travel stuff, that's precisely what Aya's documents will make her: a legal adult.
Right, I don't really know if such a thing is officially disallowed in the US, but I've known plenty of adults who have gotten their GED (General Education Diploma, a high-school diploma substitute) by means of adult education/night school.
There are a lot of reasons why that's a better option for older students (not least because most adults have to earn a living) and also why having much older students in a regular high school might be sub-optimal for everybody.
Now, a time-traveled-nominally-21-but-physically-and-mentally-still-a-middle-schooler would be an exceptional case, to be sure.