Amtrak travels to Washington, Hermann, and Jeff City, which are near the Marthasville, McKittrick, & N Jefferson trailheads, respectively (it also goes to Sedalia, but that's beyond the section of trail you mentioned). Unfortunately, the train stations are on the south side of the river and the Katy Trail is on north side of the river, and you won't want to attempt any of those bridges by bike - some experienced cyclists tackle the Hermann or Jeff City bridges, but it can be harrowing with cars rushing past just inches away.
Which brings up shuttles: the shuttle services carry people and/or bikes from point A to Point B along the trail. As far as I know, they will go to the Amtrak stations, so they could probably get you across the bridge. Or if you decide the train is too much hassle, a shuttle service can simply pick you up wherever you end your ride, and drive you & the bikes back to wherever you left your car at the start. This link has a list of the shuttle services along the Katy Trail
. Honestly I don't know the cost; but if you contact them I'm sure they'll give you an idea.
I also don't know the cost of Amtrak, but this FAQ question has Amtrak's 800 number, and a link to their website
so you can contact them for details. You'll need advance reservations, because fall is a busy time for this Amtrak route. Especially October - Hermann hosts a big Oktoberfest and tourists love to ride the train there from St Louis & Kansas City. And like you said, there are a limited number of bike spaces available.
If all the planning gets to be too much, another option is to use a tour service, which arranges all the transportation and lodging issues for you. It's a more expensive option, but it frees you to just ride and not worry about the details. Right now I'm only aware of one touring company on the Katy Trail
Hope this helps.