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Given your list of requirements and desires for this trip, frequent breaks from driving, scenic, and avoiding city traffic, I think the clear winner among the three basic routes you list is a combination of the middle one, "I-64", and the southern one, "I-81" (although you'll never see I-81) Such a route avoids Columbus, Indianapolis and St. Louis while taking you through scenic parts of Maryland, West Virginia, Kentucky and Missouri.
In this hybrid suggestion you'd leave the DC area on I-270/I-70 west to Hancock MD where you switch over to I-68 west. Follow that to Morgantown WV and I-79 south. At Charleston WV take I-64 west. Note that there will be a short stretch of I-77 linking I-79 and I-64. Continue on I-64 to Lexington KY. You will have to leave I-64 and go 'through' Lexington, but it's all on four-lane divided highway: Exit I-64 onto US-68 west to KY-4 north/west around the city center to US-60 west. Just after leaving Lexington, you'll want to get on the Bluegrass Parkway west to Elizabethtown. There use I-65 south for a short stretch and take the Western Kentucky Parkway west. Near Nortonville the Western Kentucky Parkway joins I-69 south which subsequently joins I-24 west. At Paducah, leave I-24 for US-60 west and follow that the rest of the way to Springfield.
The above route is within a few dozen miles of the shortest possible route and is just a much more pleasant drive. At just under 1,100 miles it could be done in two days. With plenty of stops, say every two to three hours, you should still be able to make it in two days, but that would require much closer to 10 hours a day actually 'in the saddle' rather than the seven you are envisioning. I think you and the toddler would be happier making two overnight stops, but it's your call. If you stop just once, Lexington KY is roughly half way. If you make two stops, then Charleston WV and Central City KY would be more like it.
AZBuck