Freezing temps pale in comparison to other dangers...
I suspect you really aren't taking our recommendations and suggestions about driving distances seriously enough. If you're trying to drive from Paducah to Estes Park in 1 day, you will be putting your lives, and the lives of everyone else on the road in danger. You simply can't drive from more than 1,000 miles - well over 20 hours on the road - and expect to get there safely, much less be in any condition to continue driving hundreds of miles the next day.
For that matter, if you only have 4 days, you really don't have time to even detour as far off the interstate as Estes Park. Trying to drive more than about 600 miles a day, even with multiple drivers, very much puts you in the danger zone - which is also the limit for professional drivers. Frankly, even that distance is more than we recommend on trips with children. The stopping points we've previously suggested are already at the absolute maximums you should be looking at for a single day on the road, so unless you can get more time for your drive west, you really have to treat this portion of a trip like a job - focused on getting there safely as your first priority.
KOA is not really a place I will ever consider when camping, the vast majority of the time, they are way too expensive and too focused on RVs. I'd recommend trying to find state parks, national forest campgrounds, or other city/county parks with camping in the area you'll be traveling. It takes a little more work to find these places, but is usually well worth it.
That's still too much when camping with a child.
Over and above Michael's well worded comments.... if you are driving even 500 miles per day with a child and have to set up camp and pack up camp each day, you are going to be exhausted before you have even got a fraction into your trip. The two year old deserves more consideration than you are presently planning. (Most of us here have travelled with small children and are only too aware of their needs over a multi day roadtrip.)
Even following the advice above, is far too much for an enjoyable camping trip with your family.
Lifey