I climbed Granite in late June of 1994. That was a low snow year and the spring temperatures had been warm. I wore heavy-ish approach boots and strapped crampons to them (we went up Huckleberry Creek and ascended Granite Glacier). The crampons were strap-on rather than step-in, so I could get away without needing my plastic boots. I did need the crampons, but not the ice axe - nor did we rope up after the snow bridge. We crossed over the top of Granite Glacier to Froze-to-Death and could kick steps later in the day.
As to Steve's comment about snow: this year, Montana is running about 110 - 115% of normal (first time in many years) although I don't know specifics about the Beartooths. I would recommend coming fully prepared for a big snowpack, just in case.