My husband had scarlet fever and his symptoms were a fever and flu like symptoms for two days and then a red bumpy rash all over his torso and face, hands and feet. It lasted about a week and then his hands and feet peeled. At first, they tested him for meningitis and we joked that he had Roseola (which my daughter had gotten a year prior). It took almost two weeks for us to confirm what he had.
My daughter at 10 months old had Roseola, which was several days of unexplained fever and then when the fever broke she developed a similar rash to what my husband got with it being all over her torso. It lasted about 2 days. When I took her to her pediatrician, at first they thought that she had a unrinary tract infections since the rash did not develop until after the fever broke.
The reason I mention both is that both infections are caused by the strep virus, which for did not show up in throat cultures. It appears that anyone can get strep in different parts of their body and develop these other diseases. We had had a nurse at a Quick Clinic tell us husband had strep and did not believe her because it was not causing him a sore throat. After some research, I found out that she was correct because there are several forms of strep and it comes out different symptomology.
Both of these diseases appeared to look like heat rash. Both were pretty much harmless because it was caught early and treated. I don't believe my daughter even had to have an antibiotic.
Here is what I would do. If your child did not have unexplained fever symptomology before or during the rash, I would go get a second opinion. In fact, I would make an appointment with my pediatrician anyway for him/her to follow-up on what Urgent Care prescribed my child. I definitely would ask my pediatrician about the fact that the rash somes and goes with baby being hot. I suspect he/she will either confirm your suspicion that it is only heat rash, or will tell you it is due to the scarlet fever being untreated and that the heat brings it out. I would not stop the antibiotics until you get definite answer from your pediatrician than what the Urgent Care gave you as you do not want things to get worse.
I hope this helps. Good luck!