Dear Friend
I did both the Beartooth and Going to the Sun last year.
Most people stop at West Glacier and head from there to the Going to the Sun road heading East.
I would suggest that you instead sleep in East Glacier the day before riding the Going to the Sun, as there is less traffic from East to West and you won't be stuck behind a RV or trailer.
Half way to West Glacier there is a visitor center and you can park and walk to one of the snow fields. You can check some photos in:
https://goo.gl/photos/RhApEB4gzKasqzUS9Concerning the Beartooth Pass, pay attention that the weather up there is unpredictable and even in summer may happen some snow. I checked the weather forecast before riding and timed it perfectly to leave the summit 30 minutes before a storm.
:-)
Cheers
Savago
Few yrs ago BT pass was 40 deg. Last yr when we rode it, it was nice. We did the entire pass with her wearing a short sleeve shirt.
The route we are taking thru Idaho kinda puts us at west Glacier. We live in Indiana and it would be a lot of back tracking to go east to west and back east again. Only have a few days to get home after the trip. Be cutting it close as it is. The park does offer a tour shuttle which is a option, esp if it rains.