RVS
Harvey Young is a neigborhood airport that's really not set up for transients. I think you'd have a hard time finding a place to park, much less get a car.
However, compared to Dallas, EVERYPLACE in Tulsa is close. I'd suggest Jones "Riverside" (RVS) and check in at the Christenson Jet Center. If they don't have a courtesy car, they can get you a rental. Of course I may be biased since I used to be based at Riverside and still consider it the greatest GA airport I've ever seen.
I have to admit, though, you could almost walk to Barrett's from the big airport. The GA side of "International" is widely separated from the commercial side, so it's almost like a separate airport, and you would have to deal with a tower at Riverside anyway. It's just that the Riverside folks are a lot friendlier (and cheaper) and you're more likely to get a car at Christensen than Millionaire.