I remember the first time you posted. I agree that your what you described is heat stroke. It can come on pretty quickly. It really takes the energy out of her. She may even vomit when this happens. She may be extremely weak even once she has had a chance to rest.
I could also be that since she is not used to this heat, she is even more sensitive to it than a native Texan.
Today here in Austin it was extremely humid. I knew I would be running a lot of errands so I was prepared with a huge cup of ice water for the air conditioned car. I also knew there was no way I could really do anything out side, so everywhere I went, I went inside immediately. This was not a day for me to do anything physically outside, unless it was swimming,. Even then, I would have gone to a pool with a shaded area.
I did not go swimming, instead after 3 hours of errands, I came home, cooled off and then took a 3 hour nap. I am still a little shaky.
Your daughter is small so she can overheat even quicker. You say she does not perspire? Or she Perspires very little? This is usually a way the body helps to cool itself off. Maybe she needs a spray bottle to mist herself, even one with a fan on it.
Before she over heats or if she is over heating, have some wash cloths that can be wet with cool/cold water and place them behind her neck, in the crease of her arms and behind her knees. This will help cool her off very quickly. We keep a cooler full of ice in the car all of the time in the summer.
Make sure her hat is well ventilated. Have her take off her hat in shady places and spray and fan her head.
Also ask her doctor or a pharmacist about salt tablets. They used to give them out a summer camps to help our bodies hold in more water.
I am so sorry she is suffering with this heat. Just give her time and she will probably become more used to the heat and humidity. It is just going to take a little time.