I would take a connecting flight. Immigration at DFW usually goes pretty smoothly - especially if you have Global Entry. I'd suggest a flight 2-3 hours after your scheduled arrival. I wouldn't worry too much about reliability of the flights as there are flights DFW-AUS every hour or so. Returning, I'd give yourself about an hour and a half if no checked bags and 2.5 if you're checking bags - bags at DFW are pretty slow and you'd have to re-check on separate tickets.

A car service, I think, would be quite time-consuming given the significant traffic that can often exist between the two cities. Your arrival date is a holiday weekend as well, and in my experience that just means even more traffic. A slightly cheaper option, if you are set on ground transportation, would be the Vonlane luxury bus. You can Uber or take the DART train to the pickup point, which is the Doubletree by Love Field (about 30 mins from DFW). The bus will take you to the Hyatt Regency across the river from downtown Austin, from which you could then make your way to the Marriott. However, I think this would be a bit inconvenient for someone departing DFW.