I recall that there are downtown offices for Avis and Budget near the main Denver station; Hertz may also have a downtown office. I'd call and see what is out there. They are going to be a bit of a trip from the temporary station, so you're probably looking at a cab to get there. The agencies were (last time I did this) only open M - F, and Saturday morning. You can drop outside of hours using a drop box.
Another option, RTD has a main station downtown, from which there are express busses to the airport. Might want to check on that if the downtown option does not work out. Tons of rental options at Denver International.
Have a great trip!