Tamiflu is for Flu.
Your child is not diagnosed with Flu. Formally. It is just a guess.
Look online, to see what age(s) Tamiflu is for.
Your Doc said its okay, for a 4 month old baby. I don't know.
ALSO, it could just be a result of her getting her shots.
Which temperature increase, can happen.
Did they give you a 'fact sheet' or FAQ sheet, on the shots she got??? At our Pediatrician, we get handouts for each shot, my kids get. For information.
ALSO, a cough/mucus can be due to many different illnesses.
Did the Ped consider that??????
Get a 2nd opinion.
Your baby is so young.
Lung illnesses can be colds, viruses, RSV, Pneumonia, Bronchitis, Croup, Pertussis etc.
They can do a Flu test or a Strep Test.
IF your baby is getting worse, take her to the ER or Urgent Care center or back to the Doctor or another Ped.
As I said, your baby is young.