DavePM's suggestion is excellent. I did most of the Katy Trail (Clinton to Dutzow) last summer and recommend it highly. It's a nice, peaceful and fairly scenic way to cross Missouri. Get a copy of the "Katy Trail Guidebook" by Brett Dufur before you go. When you get to Clinton you can pitch your tent at the big Rec Center right off the Trail. From there you can go straight east to Le Cigne KS where there is a fine park and campground at the lake. Cut down to Iola, go west on 54 and you will hit the Transamerica Trail a bit east of Eureka.
I bicycled through western Kentucky returning home to Florida and was never chased by dogs. In fact in my 20,000-25,000 miles around the country over the past few years I have never been seriously menaced by a dog and, of course, never bitten.