My doctor does not do them, unless it is absolutely necessary. This is a good source of picking up bacterias like Pseudomonas and other bacterias, plus other risks associated with the procedure. And for some doctors (not saying yours), it a huge source of income.
My doctor first tries to find other sources of a continual cough first. She looks at many signs to determine if there seems to be signs of inflammation. Blood tests, skin testing for older children and adults. Other signs include sore throats, red and swollen eardrums and nasal passages. Frequently when these are irritated, the lungs may be also.
My doctor educates her patients to remove all allergy triggers such as products with fragrances, harsh, chemicals, advises the removal of carpets and pets must be kept out of the bedroom, eat foods without chemicals, preservatives, dyes, etc. She asks about anything new in living arrangements and prescribes an air filter and hepa or similar vacuum. She asks about lifestyle, and any other possible environmental exposures.
The first thing her patients have to do is replace cleaning products with baking soda and vinegar. Change all soaps, shampoos, and personal products to non-fragrance for all family members, not only the child. My doctor has her patients on all appropriate asthma medicines, prescribes nasal rinses (you'd have to ask how other parents do this with such young kids/plus ask your doctors) plus appropriate supplements like fish oil and probiotics for example.
So often, allergies, irritants, and chemical sensitivities can cause inflammation in the nasal passages which causes a post nasal drip (which can sometimes also be seen by a doctor) although a lot of this occurs at night when lying down with less swallowing.
I am not saying whether you should do this or not do this. But if it doesn't sit right with you, I would get a second opinion. If your child DOES have a bronchoscopy and it shows nothing, you will be back here anyway. That is: how do you get rid of a cough when antibiotics aren't working? And I would also ask, "What percentage of bronchoscopes yield result in a positive culture?"