If I understand correctly, your baby is ten months old now and has had this cough for 5-6 months. Your concern is appropriate. This persistent cough is definitely abnormal and the cause of it should be actively sought by his doctors. The explanation that his cough is “because of the weather” is unacceptable, especially in the absence of blood tests and X-rays. You are
correctCorrect (new formula). You should not have to administer cough syrup, continuously. I do not know what your options for medical care are, in Switzerland, but if at all possible you should seek to have him evaluated by a pediatric lung specialist (Pulmonologist) and, as best as you can, do not accept NO for an answer. You have a right to have your baby carefully evaluated. If it is not possible for you to arrange for consultation with a specialist, at least seek a 2nd opinion with a Pediatrician, not associated with the 3 doctors you mention.
There are many possible causes of his cough, some more serious than others, but whatever the cause, it must be identified and treated, so as to reduce the risk of permanent lung damage.
Good luck