...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.
Just looking at the map provided indicates quite a bit of unfinished routing between Jasper and Vancouver.
If the miles are added up we get something around 6000 miles.
We're ready to take a train or a ferry at some point during the trip, but I still don't know if this is a realistic project as a whole.
6,000 miles (which is nearly 10,000 km) is a pretty ambitious tour, especially if you have 3 1/2 months. If you rode every single day from June 1st to September 15th (106 days) you'd be looking at a mileage of 56.6 miles a day. That doesn't factor in any days off for doing other stuff, sightseeing (which there will be quite a bit of going through the Canadian Rockies), emergency/mechanical issues/unexpected stuff, or just plain break time. Some people can and do ride 60 miles a day, every day, for months on end. Some people can ride even more miles than that on a daily basis. But not everyone does.