I don’t know how long it takes to drive from Austin-San Antonio but it sounds as if the cake will be out of refrigeration longer than usual. If you have an SUV, you might be able to pack it in such a way as to minimize the effect of the heat. A friend does this when she drives things from San Diego to LA, and doesn’t have the refrigerated truck available. I’ve been in some cars with such powerful AC that you feel as if you’re in a fridge! If your AC isn’t that strong, this might help.
Put the tiered cake in a box (I get them at the UPS store) and get a box that is larger than the cake’s box. For example, if you are doing a 12/9/6, and you use a 16” round board, you would get a 16x16x16 box from UPS. You would also get a 20x20x20 box to put the 16” in. My friend uses a garment bag box as the outside box because she’s got a delivery van and has the height to do this.
Put the cake box in the larger box and use ice packs (in baggies) or dry ice (packaged appropriately) in the larger box - you can tape it to the sides of the box or whatever works for you.