Drove 10 to 435 last September - good road with minimal slowdown. Using the turnpike would be several extra miles out of my best route.
As far as tolls -
Kansas Tollway Calculator - says a Class 4 vehicle - which includes a pickup towing a two axle trailer - should be $2.75 from the East Entrance to the US-59/US-40 northbound Exit 204, $3.00 to the US-59 southbound Exit 202, or $4.25 for Kansas Hwy 10 Exit 196.
Be sure you stay in the lanes with a toll booth, do not travel in one of the lanes marked for K-Tag.
BTW - back in 2012 Congress passed a law requiring toll agencies to adopt a single nationwide standard - so that a toll tag from one state/ agency would work on tollways nationwide - by 1 Oct 2016.
Obviously that hasn't yet been completed but is getting closer. My Texas TollTag (Dallas area) now works on all tollways in Texas, Oklahoma and Kansas. Might be able to use it in Florida this fall.