We are doing bay fishing/Nilgai hunt combo trips this summer. Come down and fish the laguna madre in the morning, then hunt late in the evening for a Nilgai bull.
www.piercetrophyhunts.com

The only thing I can see that you should really consider an exotic hunt for in the summer would be for axis. I only say that because a lot of ranches are set up to fill in axis hunting in the summer months so they can have hunting year around. Barasingha deer are also good to hunt in late July or August because they can have a tendency to break off a lot of their antler as soon as they come out of velvet in July. I think one should consider asking that to the particular outfitter that you want to hunt with. Not every ranch has hard antler axis year around unless they have a well established herd or are bringing them in for put and take. Generally most axis across the board(the greater majority of the bucks) will be in hard antler from April up to November, maybe even Thanksgiving. A rut period can be any time of year for the axis but a lot of ranches will have it in the summer. If that's the case with the axis on the ranch you you want to go to, mak sure you check references with that ranch on previous hunters for that time of year. Then decide on if you want to hunt them at that time or come in th fall and possibly expect to see some with broken antlers or maybe not as many to choose from. Ask the ranch if they bring in any for the fall hunters and how many they can expect to see in the fall versus the summer. Also consider that in the fall a lot of areas may have a large acorn crop at that time. This can help in the success or make it harder. Ask them how they deal with that. Are they hunting over food plots because they may help too.

The heat is just a small part of the adventure. To me there is much more to worry about when choosing to hunt the exotics than if you want to hunt in the heat or not. If it's for health reason I understand. But it is Texas. It's gonna be hot, that's just a given. Lots of water. No sugars. Ask how your outfitter or ranch deals with it to help their customers be comfortable. You could be riding in a truck safari style with the AC going. You could be in a blind with an AC. Texas is the King of year-round hunting and I promise you there is an outfitter out there that can help you handle the heat. Don't let that stop you from going after the prime time to hunt your game that your after if it's axis.