I've done that same route twice. There was still snow in the Montana/Idaho passes about May 10, if I recall, but not actively snowing that day. And it was pretty cold in Billings. After that you should be okay, but truly "warm" weather will be the exception in May.
I might suggest going via Michigan Upper Peninsula, then down to Flint, and then Port Huron/Sarnia. Avoids the whole Chicago mess. (Unless you need to be in Battle Creek). Another trip we went well south of Chicago below Rockford, which was longer but more peaceful.
