Baby Health Issues

Sorry for the long post... So my LO got sick 3 days after starting day care (2 months ago). He was 4mo old. We saw his pediatrician and was told he just has a really bad cold. About 1.5mo later he was still sick (mainly coughing--a wet cough). After trying to make multiple appts throughout that 1.5mo (doctor was always busy...), we finally snagged an appt. He started getting a rash, a low fever, and still had his "cold" symptoms. Doctor said his ears and throat looked fine and his lungs sounded clear, so he did a nasal swab. After finding Moraxella (bacteria which affects respiratory and ears), we got put on antibiotics. 10 days go by and he's still not better. Had his 6mo well this past Monday and doctor said he could either have allergies or asthma (umm...pretty big difference on how to approach so wtf is it..). We got 2 inhalers, one (flovent) I use twice a day for a month to see if that helps. And another (proair) only to use during extreme coughing fits. I was told he likely won't need to use the proair, but I've use it TWICE already (last night he couldn't sleep, coughing was keeping him up 😷). I'm at a loss. His coughing isn't getting better and he's now struggling to breathe during the coughing fits. Peditritcian is trying to rule everything out, but I feel my son is just not getting the right treatment and I just want him to feel better 😔