L.R.
Please be VERY assertive with the doctor.
The doc has already said "It's eczema" and let it go. So when you go back you may get the "She's already been seen and we told you then...." eye-rolliing attitude that some doctors have.
Be assertive. Ask them to consider and possibly test for strep; allergies; hives; other viruses; scarlet fever. Sit right there and TELL them what you want them to rule out. The fact that she is gaining weight without eating much is telling -- yes, kids do grow and gain, but if her weight gain seems sudden, that is not normal.
I cannot believe that they said eczema for a whole-body, bumpy red rash.
I also can't believe they are crediting her extreme tiredness to "fighting off a virus" unless they also tested for viral infections and gave her antiviral medication. (And if they gave her an antibiotic, change doctors now-- antibiotics do nothing for viral infections.)
Demand to be seen today and do not be afraid to be "that mom" who gets bold with them.