Best option is to get the merraige visa. You can't work with the student visa. And unless you already have a contact, I think to get a work here (that offers visa) would be difficult. You can play live 3month as tourist hop to Japan and come back move but that would be difficult with the pandemic and it is an uncertianty that would weigh on both of you. I'm a Korean my husband is a Brit programmer. We went through this route before and my advice is to look into marriage visa immediately. Its actually not that difficult and while it's comparatively fast it still takes some time. You can find my contact in the profile if you want.