I have only ridden it west to east, so I can't compare it to anything, but riding it the way I did was a fantastic experience. The entire climb was shaded for me, so that's a judgement call I guess if you care about such things. It was cool and there was no wind.
Maybe the other way is better. But if it dorks up the whole rest of the trip, and you are confident you can make it all the way to the top before the daily bike closure, I don't see a reason not to just ride west to east.
I made it to the top with plenty of time to spare, starting from a hotel in West Glacier at first light.
Don't miss the pie at the Park Cafe in St. Mary.