One of the best ways to do this is to ask your pulmonologist for a referral to an ID doc with whom they have a working relationship, so you can form a team rather than a competition between doctors. With luck, they will be in the same medical record system, so all of your history, consults and test results will be accessible to both of them.
After a "false start" with my first pulmonologist, that is how I have done it - now that I am stable off meds, the pulmo follows me and we will only bring the ID doc back in if things flare up again.
Sue