I agree with JPDM. Based on what we know about current Chinese policies, your only safe bets to get a visa are the UK and Hong Kong. Due to your travel schedule, any visa you get in the UK will be long expired before you can get to China. That leaves Hong Kong. Suggest that you change your flight routing to fly Vancouver to HKG. Allow at least 2 business days in HKG to get the visa, and using an agent there might be advisable to expedite matters and free you up for sightseeing.

While it's possible that the Chinese will change to a more lax third-country issuance policy between now and time you need a visa, I wouldn't bet on it.