Many thanks to all for good responses. October is out then!
What about May, June, September? We usually try to schedule our trips out of the biggest holiday months, which I guess are July, August in Canada?
July and August are the busiest, hottest weather and kids are out of school, but we definitely don't have a huge swing in tourism. I would suggest to aim for June, good weather, but not terribly hot. May and September are unpredictable and can be chilly and/or wet.
Try to figure out what you want to see, Canada is far too large to travel properly in a couple weeks, and intracountry flights are expensive. East coast Canada (Nova Scotia, Newfoundland) offer a laid back lifestyle, pretty nature, and unbelievably friendly people.
Central Canada (Ontario, Quebec) has the two largest cities, Toronto and Montreal, Quebec City is charming as well. Not much for scenery though, just a crap ton of trees and small lakes. Pretty at first, but not a lot of variety. It does open up the option of an East Coast US road trip. Boston, New York, Washington are reasonable close to each other and any of them are a 9 hour drive from Toronto.
Western Canada (British Colombia, Alberta) to me has the biggest appeal, though I'm ashamed to say I haven't been yet. Ocean, mountains, and Vancouver. Rent a car and drive south to Northern California.
Personally, I would rent a car instead of a bike. Distances are large, having the ability to cover a lot of ground can be very beneficial on such a short timeframe. Bike rentals are very expensive compared to car.