I just flew out of IAH a couple weeks ago on Delta.

It has a separate/central rental car facility. They have buses to take you there. When you're catching the ride back for your departing flight, they have two sets of buses. Four terminals and two terminals per bus. The facility is within the airport grounds so it's not a long bus ride. There seemed to be a good amount of buses with one pulling up to pick up people right after one leaves.

To get to the facility from JFK Blvd North, take a right on Rankin Road, then I think it's your first left, then the facility/garage is on your right.

I think if you got to the rental car facility at 5:45, you would have no problem getting to the ticket counter by 6:15. Maybe even closer to 6 if all goes well. You should check and see what the cutoff time for checking bags is.

BTW, if you have to get gas, the main exit from the beltway should be JFK Blvd. I saw a couple gas stations between the beltway and the airport.