My doctor refused to test me several times over a long period of time as symptoms worsened. I now have a pulmonologist.
You could insist that your doctor either do the mucus sputum test or refer you to a pulmonologist who will ( if you don’t already have one).
Perhaps having someone attend the appointment with you might help? If you have ongoing symptoms or a serious flare up of Bronchiectasis, it is hard to imagine you would be denied a sputum culture test, but it happens. Sometimes a second opinion is warranted. Don’t let a doctor gaslight or intimidate you if you are ill.
Wishing you luck.