Just stumbled upon this, been off the forum for a few months. As a resident of KC, I can attest to Smokehouse. One of the best for a sit down restaurant experience. My favorite is Joe's Kansas City barbeque. It's takeout in a gas station, but well worth it.
http://www.joeskc.com One can't go wrong with Arthur Bryant's but it is more like a lunch counter/diner than restaurant.
There's a couple I haven't tried yet, Hawg Jaw Que & Brew, just north of the show.
http://www.hawgjaw.comand Smokin' Guns BBQ a bit south-southeast of the park.
http://www.smokingunsbbq.comI don't know if these are takeout, diners, or restaurants, fair warning.