While I do think it would be a bad idea, it would be possible for the OP to hide a good amount of his trading income, especially if he properly maintains his privacy
Trying to use bitcoin to break the law is a bad idea in general for lots of reason.I surely hope that isn't what the OP had in mind.
I don't think it's possible to avoid taxes using bitwage. They report from what I believe, and so does your employer, as usual. This has little if anything to do with taxes.