In order to fly for a Canadian 704/705 carrier you need to have either a Permanent Resident Card or Citizenship in addition to the appropriate Canadian licenses.
End of story.
Getting a PR is equivalent to getting a US Green Card down in the States and it will take time. Recently there have been changes in the immigration policies here in Canada that make it more difficult to immigrate and gain citizenship. Look at the CIC website for info posted earlier in this thread. Really research this to see if this is something for you.
Best of luck.