If you can leave your plans fluid, then keep an eye on the NPS Glacier website for updates on GTTSR clearing and opening date. If you can't be fluid, then you might opt for trying it on the way back from SF.
Here is the history of opening dates.
You have to have a ticket to drive the road. It's probably a really good idea. When we were there in 2016, we drove it across east-->west and were only able to get into a few viewpoints, and one of them was not Logan Pass VC. We got to Apgar Village, waited an hour or two, then started back going west-->east (instead of returning via US-2). By this time, we could get into most viewpoints including LPVC, but were only able to do one of the short trails we wanted to do.
Donna