A bike is nice, but is not a must. The paddock/course layouts cut the distance down on the walk. Plus, if you are running the Pro Finale you should be able to pit closer to the courses & indoor rest facilities. Also, you could go to Wal-mart and buy a super cheap bike while in Topeka.
I've made the drive 14 times (and flown one other) and have done it in all conceivable fashions, about half of them straight through. Although, it's definitely a much more pleasant and safer drive if you stop to get some sleep and a shower. Please do avoid driving all night if you can. Getting a decent night's sleep on the way out certainly helps you for the Finale - being sleep deprived is not good for your racinig.
You should be able to make it in about 19 hours from Raleigh without pushing too hard. Leaving Tuesday night works. Also, if you leave really early Wednesday morning (e.g., 4 - 5:00 a.m.) then you have two choices depending on how you feel: 1) stop somewhere deep into Missouri and sleep so you have a short trip on Thursday morning like Pastorius said, or 2) drive all the way to Topeka, arrive between 10 p.m. - midnight CST, then sleep in late Thursday. Any of those options should work well.
We prefer going the I-77 to Charleston to I-64 to I-70 Route. It take the same time as I-40, but time seems to pass faster (to me), there are more/better places to stop, and I've rarely had traffic problems going that way.
FWIW, my record is 16:30 in the Spyder with a tire trailer, from Topeka to Hillsborough after spending all day in 107 degree temps at the Midwest Divisional - the heat was punishing. My average speed was 67.94 MPH including stops. That was the event where Tim took a shower in the carwash at South Topeka Blvd after the event before getting on his plane. He said that people in the car wash looked at him like he was crazy, but it was worth it. He never told me whether he used the lemon wax or spot free rinse features while in the car wash

I was upset that I didn't think aobut it myself, that was a long and brutal trip home. I'm too old and wimpy to do that now.
Eric