I'm assuming that you are not stopping en route from the border to Cochrane.
The route I would pick to save time would be to take highway 2 to the south side of Calgary and then take 22X (Stoney Trail) east to 201 (also Stoney Trail) and follow that all the way around Calgary to 1A (Crowchild Trail) to Cochrane. No stops, all off ramps and you avoid most of the city. The south side (22X) and the NW side of 201 is the only place you'll encounter significant traffic, even in rush hour.
The other way around Calgary that I'd recommend is a bit slower but has less traffic. Take Hwy 2 to 22x but go west towards Bragg Creek and hook up with Hwy 22 there. Take 22 straight to Cochrane.
If you have more time, Hwy 3 to 22 and north to Black Diamond is more scenic but the highway is not divided. Both this route and the one you have suggested involves getting from Black Diamond to Cochrane. This route is a royal PITA (pain in the ....) because of all the stops and turns. It is a nice road, just one that we never take with our trailer.
To put this in perspective; we store our trailer close to Cochrane and camp south of Longview during the summer. We NEVER take the trailer from Bragg Creek to Black Diamond via Hwy 22 and usually just go right through the city.
I can understand your apprehension about driving through Calgary, but you won't have any issues taking either of the two routes I've suggested because you are just skirting the outskirts of Calgary with both of them. Most of the time you can just stay in the right hand lane and take off ramps and exits.
