If you are going to stay in any of the lodges, make it Many Glaciers. The setting in magnificent. However, remember that these are really old lodges and whatever the furniture, the walls are thin. For the most part, we stay elsewhere to get a good sleep and have lunch at the various lodges to soak in the ambience.
We stayed at Many Glaciers mid-June the day or two after they opened for the season. There is still lots of snow around at that time of year which means the critters are down in the valleys. From the dinner table at Many Glaciers, overlooking the lake, we saw grizzlies on the hillside, a moose and baby swim across the lake and, of course, the ever present bighorn sheep wandering around. We then moved up to Waterton with lots more critters+babies around. Things are pretty quiet still since the season isn't up and running until the snow melts out more but it's a nice time of year.
Last year, we were at a timeshare in Whitefish in early June and thoroughly enjoyed that. It's only about a half hour from West Glacier so it's very convenient. The Flathead Valley is green with fresh growth and it was a beautiful time of the year to be there. We drove to the east side and stopped in at Two Medicine Lake where it was REALLY cold with a strong wind blowing off the ice and snow that was still around the lake. Make sure to take some extra warm coats.
Sue