Japan's fighter quandary is a big mess. While I agree that a 2-engine fighter is better suited for them, I don't know if there are any good options out there. The F-2 is long out of production and it was a single engine fighter to begin with, the F-15J has two engines but it's also long out of production. I doubt it'd be worthwhile to resurrect either of these planes.
For the other Western fighters in production, we have the Gripen (1 engine), the Rafale (2 engines), the Typhoon (2 engines), F-16V (1 engine), and F-15EX (2 engines). None of these are a good fit for Japan so I think that the F-35 would still be a better choice. I suppose that Japan might wait around for the KAAN, but that feels like a step backwards. The KF-21 might be an option as well, but it's probably too embarrassing politically. At that point, Japan might as well place an order with Shenyang!
They'll probably end up buying more F-35s as a stopgap and then the F-47 when it comes out since it was already offered to them. I don't expect Japan to ever develop their own fighter again unless they are absolutely forced to. They just don't have it in them anymore (for many many reasons). However, they could potentially make a "kai" version of the F-35 and F-47 down the line. As far as I know, they do already produce the F-35A locally (finally assembly and maybe some parts of it, I'm not sure).
GCAP might actually be ready before F-47 so it doesn't make much sense to abandon the program. And I doubt that Japan has the wherewithal to make indigenous variants of the F-35 or F-47. The former is going to be an absolute nightmare given the kind of mess the software situation is in, and you'd be nuts to think that the F-47 is going to be any better. And that would only be possible if the US gives permission to begin with.