I'd suggest swinging down thru STL on I-55 to I-44 into OKC, I-35(W) into FTW, and then getting on I-20 vs. betting on I-40.
Not a lot of scenery on I-40, but there are tons of trucks, and the weather will turn on a dime. If you've grown accustomed to driving in Chicago winter weather, then you can handle it, but chances are most the other drivers are clueless.
You shouldn't have any problems reserving the RV park at the rim in the winter, but you'll need to be self-sufficient for water/sewer. Williams is 2000 feet lower, so you'll have less snow/ice to contend with, but we like being -at- the Rim and not driving all day to/from it.
The other advantage of being there in the off season is you can drive out to Hermit's Rest (the road is bus and handicap permit only during high season).
Seeing the canyon in snow? Priceless.