There are a handful of B&Bs that explicitly say they'll allow pets:
Das Gast Haus Nadler, Defiance (mile 59)
Angels in the Attic, Hermann (3 miles off the trail from mile 101)
Harbor Haus Inn & Suites, Hermann
Les Lavandes B&B, Rhineland (mile 105)
Jefferson Inn, Jefferson City (1 mile off the trail from mile 143)
Hartsburg Inn, Hartsburg (mile 154)
Cruces' Cabooses, Windsor (mile 248)
This doesn't mean that no one else accepts pets. Some of the B&Bs want to talk to you first and find out what kind of pet you have - from your description I would imaging that many B&Bs will not have any objections to your little dog. Try calling some of the B&Bs listed on the Katy Trail B&B Guide
, who do not have "Pets: No" stated in their listings - if you ask, they just might say it's OK.
You might also have some luck with motels near the trail, which you'll find in Clinton, Sedalia, Boonville, Columbia (9 mile bike path ride from the Katy Trail), Jefferson City, and St Charles. Here's a list of all lodging on the trail
Do you have a photo of your dog riding in his carrier? If you want to send me a picture, I'll publish it on the website. Good luck with your trip.