Is the phrase "thanks, but no thanks" appropriate in an email sent to a supervisor?
It sounds extremely rude, I am afraid. I would assume mitigating circumstances for a non-native speaker, but the "no thanks" permits "thanks" to be interpreted as substantive, and thus has a highly dismissive connotation which should never be used with your superior, and neither with a friend you would like to keep.

The connotation that shines through (at least in the UK) is a sarcastic "Yeah right, you think you do me a favour? In future, please spare me your ideas."