Not sure exactly what you mean by "Northern Tier" since that usually refers to an Adventure Touring route and your proposed route is much further south. The best website for climate normals data is from the Western Regional Climate Center - click on the state, then the community. There are far more choices than most weather sites - so you can find out what it might be like in Stanley, Idaho or Yellowstone Lake.

http://www.wrcc.dri.edu/climsum.html

A word or warning - it sounds like you are from back East. The climate variations in the West are greater both in terms of temperature differences from night to day and weather changes from warm and pleasant to cold and dangerous. Even though Stanley, Idaho is well south of the Northern Tier, it is often the icebox of the lower 48. Spring comes late to the Northern Rockies. You can certainly expect snow in May - even into June. I've seen it snow on July 4th in Jackson, Wyoming. Even if you don't have snow coming down, you will have lots of snow still on the ground in the higher elevations in June since it hasn't had time to melt yet.

If you are starting in mid to late May you should expect temperatures in the low 40s and, occasionally, into the 30s. The AVERAGE low for Yellowstone Lake in June is 32 - that means if gets colder than that about half the time. If you have a 40 degree bag, depending on the make, that often means anything less than 50 is uncomfortable. I'd go 32. You should plan to take glove liners and a fleece earband. These two items take up no space and will warm youconsiderably on chilly mornings - plus you can wear them to bed at night.

Remember - in the West you should always be prepared to sit out a day or two if a late-season snowstorm is on the way. Distances between services and help can be considerable. There is no need to place yourself at risk since these late storms are short and the weather afterwards is always fabulous.