We now live in central Washington, lived in Minnesota for many years, and made this trip several times in early May. It CAN snow in April as far south as Minneapolis. If you might like data on snow days in April, here's a national resource: http://www.ncdc.noaa.gov/climatenorm...NDYFXX_fmt.dat
Two KOA campgrounds on your route (not that you would necessarily stay there) open May 1st - West Glacier, MT and Bemidji, MN.
Surely an Alaskan doesn't mind a little snow. If you're adventurous and don't mind laying over a day while they clear snow, go for it!