"I'm a perfect bilingual" only has 2 results too, btw.
"I'm a complete bilingual" gives no results because when you are a bilingual, you are a person who can use 2 languages. When complete is used as the adjective, it modifies the person, not the ability, so this is the core source of the weirdness and why the expression is not used. But it could be used in special situations like for sarcastic effect, because there's the expression "I'm a complete individual" that is used when, for example, you have met the love of your life and you feel like you are complete, and this expression itself is a kind of play on words. In conclusion, the modifier for the degree of the ability to speak two languages should be applied to the ability, and that's why the expressions involving bilingual in the sense of the person don't return many results.