When I was there in July 2012, some of the free shuttle buses had bike racks on the front of the bus. If they still do that and if the weather was bad or road was icy (unlikely), the shuttle could be an option. But, I am sure you want bragging rights to say you peddled up the mountain.
If I am reading your map correctly, you are going east to west. That is the best way to cycle for two reasons:
- only one cycling time zone restriction, that restriction is on the west end of the park and you can hang out at a general store where they sell beer waiting for the time restriction to end.
- less climbing from east to west, see elevation profile from the GPS data from my ride.
A side note, the day before I went over Going to the Sun Road was biking from Waterton Park to St Mary. The climbs on that day were the toughest during my whole trip. The elevation profile for that day looked like this.
