I've been talking to a lot of people who have done long bike trips like this and I've decided to flip my trip and start in San Francisco and make my way home from there.
I re-looked at this a week or so ago, and I don't think your original idea (Banff to SF) was nearly as crazy as some have inferred.
First, the route looks pretty reasonable. You've got two passes I'd be concerned about, coming out of Banff and Lolo Pass west of Missoula. The rest of L&C to the coast is lower, and the coast itself will be temperate barring late winter storms. Looks like the high point of the route is around 5,000 feet, so you can safely ignore those warning about 10,000 foot pass closures.
Second, using Missoula as the proxy for inland weather, looks like the average temperature is 30-45 degrees in March. It's quite possible to cycle in that kind of weather, although some people's preferences may be to wait until it's warmer.
Third, the route and roads. The roads from Eureka to Missoula have approximately zero rideable shoulders. Even if it snows, once the road is plowed, motorized traffic will have just as much room to pass you in March as they would in July. Other traffic on most of these roads is pretty light, thanks to the AC route selectors, so you shouldn't have much problem. I don't know about Canada south of Banff, Lolo Pass, or the L&C route west, but I suspect they're similar.
All that said, here are some recommendations if you (or future readers) want to try this.
- Be prepared for cold rain, cooler temperatures, and perhaps snow. Your load will be slightly heavier than a summer tourist's.
- Check the weather forecast regularly.
- Keep a financial reserve for a few nights in a motel in case a major snowstorm hits.
- Local knowledge trumps any 'net posters from afar. If someone from Seeley Lake or Lolo tells you (s)he rides roads five months a year starting April 20, pay attention. Be wary of advisors who never ride on the road "because it's too dangerous," or ride in different locations. Logan Pass closed until July is as relevant as Alabama roads never close if your route doesn't include Logan Pass or Alabama.