Don't worry about getting a ride from BRC back to Reno. You can arrange that easily enough while you are there.
I recommend you stay until Tuesday, to avoid the Monday Traffic Jam. Technically, you are supposed to leave on Monday, but I am quite sure this is not being enforced until Tuesday afternoon at the earliest. It is simply impossible for 45.000 people to get out a one-lane driveway in a single day.
And if you have a flight booked, allow many extra hours to get to the airport. Looking on a map, you might think it looks like just a two-to-three hour drive. Not so in reality -- specially on Monday.
So concentrate on getting there. I'm thinking you could simply walk thru the airplane before landing, and ask everyone-who-is-not-wearing-a-business-suit if they are going to Burning Man. You will probably find someone who is renting a car, or someone who will split a taxi ride with you to one of the ride-share spots mentioned earlier.
Oh, and to be sure: The road from Reno to BRC is largely UP-HILL -- majorly up-hill.