I am so happy to hear you have a team dealing with this - it is the best approach, especially if they are in communication with one another.
One way to produce sputum, which I needed to use during treatment, was to nebulize with 10% saline solution under supervision of the pulmonology nurse. Man, it was irritating, but I was able to produce enough for a good culture. Near the end, she would also do a little manual percussion on my back to help me along.
Sue