Not sure what your question is?
You say your 5th great-grandafather came to New France (now Canada) in the 1600 (his name would have been Beaudet or Baudet but not Beaudette). When the French arrived in "Canada" they spoke French and would have learned indian languages.
I am not sure about your question "if anyone knows how the name Beaudette got to Canada" .. do you want to know why it changed (probably because most people were illiterate and left it to the priests to spell and they spelled it like they wanted to) or do you want to know who your 5th great-grandfather was?
Let us know so we can help you find answers :-)