Cat with runny nose

My foster has had a runny nose, sneezing and a few asthma attacks since I got him (7wks). I took him to the shelter clinic twice. The first time they said he had a cat cold, to send them videos of when he has an asthma attack and it should go away within a few days (that was at week 4 of having him).

I took him back to the clinic last week because he was lethargic, diarrhea, mouth full of mucus, and one asthma attack. I sent them a video of the asthma attack and at the check up they said he had a few infected teeth. He had surgery to have them pulled and they said he's just a sneezey cat. Once back with me he was blowing snot bubbles, breathing hard and sneezing.

He is doing better now, he's on a strick wet food and warm water eating routine, his poops are now solid and he is a bit more active but it's clear he's not 100% (especially at night). I've never heard of a cat being "sneezey" before. My last cat had cat asthma so I know 100% what a cat asthma attack looks like (luckily he hasn't had one since coming back from the clinic).

He is currently on antibiotics for his teeth. My question is, have you ever heard of a sneezey cat and how do I get the shelter vet clinic to take me more seriously? I'm worried if I press too hard they will make me return him and he won't get the care he needs. Any advice?