We drive I-90 to and from WA frequently, one way at least once a year. From Post Falls, The passes are the hill just east of Coeur d'Alene, Forth of July, Lookout, Pipestone - the continental divide east of Butte and Cardwell "hill". We have done them all in both a gasser and now our DP. You should have no problem exept they will slow you down. In June you can get any weather from rain to snow or freezing ice. Just take it easy as the various DOTs do a great job of keeping I-90 open. You can bypass Cardwell hill by taking MT 2 but it is an older road and may be narrow as well as winding. It is part of the old US 10 that I drove many times years ago so it is not bad, just old design. There is a cavern, Lewis & Clark, along that route that will be open by then if you are into those.
287 from Three Forks to West Yellostone is fairly good road and well maintaned. It is the alternate route to WY from Bozeman. The truckers are discouraged from using 191 due to the winding nature of that road. 191 is very scenic as most of it is beside the river and in June will be mostly Class 4 and 5 rapids with lots of rafters etc. to distract drivers. 287 has several lttle towns along it, excellent fishing, Madison River, Quake lake and the Quake visitors center. Great story there.
Just to the north of the turn off from I-90 to 287, there is a deli that is pretty good for fresh rolls etc and pretty good lunches, soup/sandwich. That is out stop when we take the MH for its eriodic exercise .
Enjoy the trip.
