If I were doing what you are doing, I would fly into MCI and take the trail from Kansas City to St. Louis. Then hop on Amtrak in St. Louis to return to KC and ride back to MCI. Amtrak has roll-on service between the two cities. Additionally, you will be going slightly downhill overall. Plus STL is a bit more of an exciting destination than KC. If it is easier to fly into STL, then you can do the reverse. Or maybe the departure time of one direction of Amtrak is better than the other. Of course, it may be cheaper to get a rental car on one or both of the legs to/from home or even drive your car to one end and take the train to the other.
Anyway, I have some draft routes (researched but not ridden) and I can send you the GPX files.
The KC portion has too many turns to give a summary here other than to say from the end of the Rock Island Trail to downtown KC, MO then up to MCI. From Machens, the route I have takes county roads to Afton via Clark Bridge and then the bike paths into St. Louis to the Arch. From there, you are on your own for the ~1.5 miles to the Amtrak station.
If you want the files, send me a private message.
Also, you should be aware that the Katy Trail can be very humid in the summertime. It is not an especially pleasant ride when the temp and humidity are the same.
Tailwinds, John