Depending on the state/jurisdiction, it may clear in the system immediately.

When I lived in the US, I had lost [misplaced] my North Carolina license and needed a replacement, but due to an unpaid ticket in California, and a suspension of my driving privilege there, I could not receive a replacement until it was paid and the suspension lifted. Once I paid it, I was able to order my replacement license the next day. Now since it was an out of state ticket, my license was not also suspended (just my privilege to drive in CA) so I don't know if the turn around period will be as quickly for you, or even if the suspending state has the same system in place.

As far as running the driving history, it is something that Hertz has the capacity and authorization to do, but it is not something they do every time. It is possible they will not, and you will be fine just showing your license and renting a car. However, there is also a chance that you will be randomly selected for a driving history check. So as also suggested in this thread, it might be a wise decision to have a backup plan.