It has definitely been done before, many times, but of course requires quite a bit of planning. As JD pointed out, many fly-in riders bring their own bikes, so at least they don't have to worry about how to get to a rental place and how to return the rental bike. But of course there are plenty of other issues to deal with when transporting a bike by air.
You may find it much easier to fly into St Louis rather than KC. The St Louis airport is less than 10 miles from the Katy Trail, whereas KCI is about 100 miles from the trail. Also Amtrak is pretty easy to reach from the St Louis airport, via "Metrolink" (which may or may not matter to you, depending on how rail figures into your plans).
One idea: Katy Bike Rental in Defiance, which is on the St Louis end of the trail, does both bike rental AND shuttling, so they might be able to help you out. Suggest you talk to them anyway just to help you figure out what your most feasible options are (this time of year the shop might be closed on weekdays): 636-987-2673
If you are considering the St Louis airport, be sure to look at this web page (which also includes a shuttle listing):
getting from St Louis Airport to the Katy Trail.
Sure it takes a lot of work and planning to pull off a trip like this, but that's part of the adventure, right? Good luck!