Your concern is that a breach of the site's data may leak some information about you that you wouldn't want to leak, yes?
If so, and if you can still use similar methods to navigate the site in question, use those methods to edit your account/profile details to scrub the account of anything that you wouldn't want to leak. Change it to use a fake name. Change the email address to somthrowaway email address. Change the password to something unrelated to any passwords you could possibly use on any other sites so that if the hash is leaked and brute forced, no one can use that to gain access to any of your other accounts. Delete individual posts or pieces of content that you've uploaded.
Actually, I can read (barely) enough Japanese to figure out that the registration process seems to only want your email address and password. (Though I haven't gone through the whole signup process.) You mentioned uploading a file, yeah? I'm guessing the amount of stuff you'd have to do to overwrite/delete every bit of data they have on you is pretty limited.
And, yes, I suppose there's the potential caveat that that might not affect backups and such, but I'd wager a lot of the other account deletion requests you've done don't affect things like backups either.