> In Japanese, he or she is called バッチリ日本人. (^_^)

I'm only a beginner in Japanese so I might be wrong, but I don't think that's correct. (Or am I missing a joke?)

バッチリ means "perfect" or "fully prepared" and it's an adverb; 日本人 means "a Japanese person". I can't find anywhere stating this phrase means an illeist.

One way to say that is just 一人称を名前で呼ぶ人, which means "a person who uses their name as their first person pronoun."