Just a heads up (you may know this already):
Mid-June is still very wintery on the Beartooth Plateau. Snow will be abundant (and not necessarily well consolidated), creeks running high much of the month (but perhaps lots of snow bridges?). Snow storms with significant accumulation are not uncommon. High temps may not crest 40°F, lows could drop below 20°F or lower at high elevations.
More importantly, it is possible that the road to the trailhead will not be passable at that time of year (it tops out at 8800ft). Check with the USFS folks on that one. Even one good sized drift will shut down that road for a while. On the other hand, it is a motortoy playground, so someone might break trail. But you may be approaching from the highway in mid-June.
The Fisher Creek SNOTEL
site is a stone's throw from the trailhead and should give you a good idea of snow conditions as well. Currently, 100" of snow on the ground. Last year on April 6th 2011 we had 135" (a better snow year). The last snow melted off on July 18th 2011 (there was still 58" on July 1st 2011) 2011 was unusual though. Most years it looks like the snow is gone around July 1st at that site.