I'm mapping out our route south, so I had my stuff out so I thought I would take a fast look for you. From what I can tell you got real slim pickings in that area. The reviews of the few private campgrounds in the vicinity were all very poor and down right concerning. No casino's, state parks, or city/county parks to choose from.
There is a Wal-Mart and Cracker Barrel in St. Robert right off of I44. Might not be the preferred places, but could be worth a call to see if overnighting is allowed. Appears to be ez on/off and for one night might be OK. You could always drive thru them, and if not acceptable keep heading down the road.
ON EDIT:
Glad now I took some time out for your post. Turns out I'm having the exact same problem on my route in one spot and forgot all about Cracker Barrel. Seems to be my best choice for my problem.
For you the one I'm talking about is (from their website):
St. Robert
I-44 , Exit# 161
I-44 & SR Y
128 St. Roberts Blvd.
St. Robert, MO 65584-3311
Located off I-44
* Bus & RV Parking Available
(573) 336-7273