cough for 4 month old

my lo has been coughing for a few weeks. it started out with runny nose first week, no see doctor. 2nd week went see doc then no lung infection, given nose drops. stopped runny nose and chesty cough in day . 3rd week, only at night now has continously cough. but feed milk then will stop. if never feed milk will keep coughing and sucking hands but still able to sleep. Doctor say is too long and need antibiotics if 2nd week didn't cure. but I feel cough is just ok, bb goes infant care anyways and hard to cure fully. ifc say if antibiotics, she will likely have watery stools and will be sent home. should I still bring her to see doc and get antibiotics for the 3rd week? I have tried all cough patch, onion patch, garlic balm, lemon bath, chest rubs. all no use 🫠 to me I don't mind the cough and doctor keep saying lungs are clear and she seems generally happy.