We've travelled the routes through the Canadian Rockies quite a bit, both with a car and towing a trailer.
Hwy 3 is a very good route. It is a bit slow through the Crowsnest Pass area, but you are towing a trailer so it is not that big of a deal.
The other option would be Hwy 93 to Radium and then south to Cranbrook. The section around Radium is a bit steeper but it is fine too.
I would avoid the TransCanada around Golden if you can. It's not too bad, but the other two routes are better.