I had something similar but they gave my LO antibiotics in the hospital. I would go back to your gp
We had this mine was poorly 5 days temp of 40c.said it was tonsillitis so had antibiotics 4 days in still the same but blue hands and just worse I struggled to see w doctor the receptionist made out I was crazy doctor seen him and sent us to a&e he tested positive for influenza A. Definitely go back ask for a chest x-ray that was the bit that the doctor said he made a strange sound so wanted one and it showed something in his lung. We were kept in and the test came back positive hours later x
I would go back to the dr and demand they figure out what's wrong with him, they should do bloods.