Can I visit Beijing if my passport expires in 5 months?

I have a trip to Beijing planned (flights already booked) in 4 and a half weeks. My passport expires ~5 months from today. I am Canadian but living in New York, and to renew my passport, I must mail it back to Canada, with a processing time of 20 business days. If I do this, I should receive my passport before my trip, but I will not have enough time to get a Chinese tourist visa. Is it possible to get it first THEN mail my passport back to Canada for renewal (the timing should work out if this is possible)? Is it possible to get a Chinese tourist L if my passport is not valid for the next 6 months? From my understanding, this requirement is prevent issues at customs when entering China.