2 year male breathing issues

I recently adopted this 2yo neutered male with FHV. He was at the shelter for over a month and underwent multiple antibiotic treatments for presumed infection, but after it wouldn’t clear they did rhinoscope to look for foreign body. He ended up being diagnosed with FHV after all test were negative.

He has lots of boogers and sneezes then all over the place which has improved since he’s settled in over the months. The thing I’m concerned about his he has episodes where he puts his head low and tilts his head back and does this cough/exhale maneuver for ~30 seconds. Before and after he is completely fine and goes back to sleep. To me, this seems like asthma, but he’s seen multiple vets (shelter and post adoption examination) who say it’s the FHV. This is my first FHV cat and I’m wondering is this common/expected or something I should pursue further?

Thanks everyone.