If you can delay the trip, that would be ideal. Typically, all but a few roads are open by the 4th, and most in late June. This year was on a substantially later schedule, though, due to the snowpack. It's awesome to be on them when the roads first open so you get to see all the snow, but I would suggest waiting a bit longer to make sure they're all open by the time you come out.

Late July would be best, or specifically, around the 20th. Generally across the state, wildflowers are at their peak roughly between the 15th and the end of the month. Might as well see them if you can take off during that time. Plus, that means you also get away from the humidity of MO during the heat of summer! Sounds like a no-brainer to me!