Wes, if you don't find a bike-rental shop at Nuits St. Georges and you don't want to bike from Beaune, the bike shop in Beaune will deliver to Nuits (for a fee). You should correspond with Florent at Bourgogne Randonees; even if you don't rent from him he will have good advice.
I have only ridden as far north from Beaune as Echevrone. That stretch was hilly, but not extremely so, given the low gearing choices on those bikes. Still the answer might depend on your comfort with hills and also what else you are carrying. (Is the the alternative to your $200 auto rental?)
The ride to Nuits St. Georges from Beaune, btw, is flat, so you might choose to go that way.
A bicycle is a spectacular way to tour Burgundy and can be an alternative to a car. However, you may encounter inclement weather, and it is probably a good idea to have either the right gear, a plan B, or both.