...go to Ottawa, ON on the Trans Canada Trail (300 miles)...and finish with the Trans Canada Trail again that will take us to Vancouver (720 miles)...
The Trans Canada Trail is more concept than reality at this point, as there are still many sections that need to be built. Before committing yourself to any of its routing, make sure you do your research. http://tctrail.ca/explore-the-trail/ Just looking at the map provided indicates quite a bit of unfinished routing between Jasper and Vancouver.
I wouldn't exactly call the Trans Canada Trail more concept than reality, much of it does exist (73%), some links are still unofficial or under negotiation for better routing. The intention is to complete the remaining links by 2017 for Canada's 150th birthday. However, the entire trail was never conceived to be specifically for one mode of travel and many sections of the trail are not for bicycles, for example some of them are canoe routes through Northern Ontario. Where the idea of the trail is misunderstood is it's not supposed to be something you do in its entirety in one shot — the "concept" is that it is one contiguous recreational facility that you share with others who are also on it at the same time all across the country. What can I say, it's a soft, emotional Canadian thing
There are several cycling routes from Montreal to Ottawa, the TCT is certainly the most circuitous but quite scenic and rustic, winding through the Laurentian Mountains. Make sure your tires are adequate for gravel, the section into Ottawa might not be as well travelled and tamped down as the "Petit Train du Nord". There are parallel secondary highways as alternatives though. Between Ottawa and Kingston, there is a missing link in the trail from Carleton Place to Smith's Falls — a local group is negotiating an abandoned railway, they just pulled out the tracks last year (I'll try riding it this summer with a mountain bike to see what it looks like). If you think you'll make it to Ottawa before the first week of July let me know.
As for Jasper to Vancouver, one option you could consider is to ride west along the Yellowhead Highway to Prince Rupert. From there, you can take the ferry to Port Hardy on the northern tip of Vancouver Island. The ferry ride is fantastic, you'll see whales as the ship weaves around the tight passages between the snow capped coastal mountains and islands. There is a ferry to the city of Vancouver from Nanaimo, or you could go down a bit further towards Victoria for other connections. BTW, I'm actually planning to be on the Yellowhead Highway in early July.