Do the sensitivities on the culture show widespread resistance to antibiotics? That maybe reason for doc’s recommendation. If it’s sensitive to other antibiotics, you might ask for a longer course of one.
My only experience with pseudomonas in sputum was with p. putida, a less common species, that I produced after I’d been sick with a sinus infection and cough. PMD had treated sinus infection with amoxicillin. My pulmonologist later suggested a 4 week course of nebulized tobramycin. We ended up agreeing on watchful waiting because the pseudomonas numbers were low (and his suggestion caught me off guard).

Is yours resistant to Tobramycin? Note- not everyone can take it. I’m just throwing out an idea.

Two things. I don’t have MAC complicating things.
My pulmonologist recently had me start nebbing 7% saline, hoping to decrease the number of respiratory infections I get this winter. I’m optimistic that it’s working because young grandson became very ill recently just hours after I’d kept him and I didn’t get it! Last winter, I caught everything. Are you nebbing with hypertonic saline?

Good luck to you. These decisions are tough..