"The Range at Austin" located down on I-35 @ Slaughter Lane on has a limited selection of full auto rentals.
Last time I was there they were pretty much just rented subguns and you will obviously be indoors since it is an indoor range. The Range at Austin is pretty impressive from a general layout perspective as they "spared no expense" when they built that place. Its also going to be the closest to downtown and is open until later in the evening.
Best of the West up in Liberty Hill used to have a guy who ran an outfit called "Ballistic Ops" who rented machineguns. Ballistic Ops has a facebook page and would be at a folding table near the check in desk at Best of the West.
I have not seen the Ballistic Ops guy in a while so I am not sure if he is still there doing MG rentals or not but I would call Best of the West to confirm before driving out to Liberty Hill. BOTW is an outdoor range so that may also be more your vibe. The handful of times I crossed paths with the Ballistic Ops guy he had a small selection of subguns and some smaller/intermediate rifle caliber machineguns (M4, etc.)
There is also Sanborn Shooters who rent machineguns at their local range in Smithville and OX Ranch who rents machineguns and tanks out in Uvalde. Both of these folks are a farther drive but it gets you into the belt fed and larger class of firearms.
Good luck and hope this helps.