Very funny Brian, that was the Bitterroot river, but ya, not a great idea.
I've floated the Missouri only about a half dozen times but can tell you that early May would be a great time to go. There are no crowds at that time of year. You may get some awesome dry fly fishing to large pods of sipping trout.
The river below Craig has it's flow rate regulated by the dam at Holter. If your down there early enough in May you be able to still see the schools of large Rainbows up at Holter Lake above the dam.
The great part about the Missouri @ Craig is there is a road on both sides of the river all the way up to the dam which allows for some easy walk and wade access. There is a nice pub in Craig and within walking distance to the campground, river, fly shops. They are open for breakfast, lunch and dinner.
Once you get into Craig you could head to the "Trout Shop", you will be able to get some the river report and pick up some hot flies. The owner loves Canadian beer you can't get in Montana, maybe you could bribe him for some secrets.
Hopefully, I can make it down there this May too.
Good luck.