No exceeding speed limits?
You say "not by exceeding speed limits too much?" Like staying within 10 mph?
I don't have adequate software at my fingertips right now to determine the shortest/best route, but it looks like I-80 to I-76 to I-70 to I-15, or use I-70, I-44, I-40.
My goal would be to stay at a speed that's R&P -- even if a bit above posted. If you stay within 12 mph, you MAY escape attention (less than 10 over, you almost surely will). The difference can be significant over a long distance if you can maintain a steady speed and minimize stops.
Also, keep in mind that patrol coverage and traffic are less at night. Both of these can make a trip like this less "complicated." You HAVE to maintain a steady speed -- in doing so, you do not have to drive dangerously. Also remember that a traffic stop is 20 minutes, minimum, on the side. That'll really mess with your average... :)
Remember also that the faster you drive, the more stress and fatigue you develop. That's a fact. So plan your speed carefully and hold it steady as much as possible. Don't forget to investigate your route in advance for road construction -- choose the route that has the best combination of brevity and less work zones. Also, optimize your stops -- plan for gas, food, & rest all at once.
After all of that, NO trip is worth dying for, and you don't have the right to endanger others. So keep your head on straight and plan/execute this so you do not do either. Win this "dare" with your head, not your foot! Good luck. Bob