A few years ago when I was touring in Alberta, I met a young man from Japan at the Lake Louise Hostel. He had cycled from Vancouver to Jasper and then down the Columbia Icefields Parkway to Lake Louise on a BMX carrying a huge backpack. He had plans to cycle across Canada but when I saw him, he was suffering big time. I don't know whether he completed the trip or not.

So, anything is doable if you have the strength and will to follow your passion. Personally, I would stay with the comfort of a larger bike with panniers or trailer. Good luck!!