For a lot of years, usually the entirety of Going to the Sun Road doesn't open till mid or late June. Though the opening date of the middle section of this road near Logan's Pass(the last section to be shoveled off of snow), varies each year. To the OP, I would suspect your whole group of 6 wouldn't have any issues being able to be seated near each other in coach. Especially since you'd be leaving on a Saturday(thanks computer calendar), when a lot of Amtrak routes have fewer riders vs. trains on Fridays and Sundays.
Wouldn't be surprised if a significant amount of riders got off at any of the stops near Glacier, before your group boards the train at East Glacier. BTW when you're at East Glacier waiting for the train to arrive at that station house, I did really LOVE how there were a lot of brochures for local places in the brochure rack! Even found a few for Canada(specially places in Alberta), of all places! Also be sure to check out the Great Northern Railroad history boards that are left inside the East Glacier Amtrak station house to this day, and also inside the Glacier Park Lodge north of the Amtrak station. Remember that before the train arrives, that there is a free shuttle that will drive you from Glacier Park Lodge's front door right to the train station building.