The symptoms you desribe fit.
The skin clearing when you cut it from your diet, and the extra fussiness and coughing when you reintroduced, would be enough to confirm it for me.
The fussiness is most likely your baby being in discomfort/pain.
When you cut it out you should see improvements in the first 2 weeks, a bit more by 6 weeks, but it can take months for the gut to heal and settle properly, so will be longer for all the symptoms to resolve.
If you feel you need to be more sure, the recommended way to do a challenge is to keep breastfeeding baby as normal cut dairy from your diet completely for at least 2 weeks, then have either a full glass of diary milk or adult size dairy yogurt. You then watch and wait for 72 hrs. Monitoring for any sign of reaction in baby.
Your GP can refer you to dietician.
Re weaning, we used oat milk instead of diary. The protiens in Soya are similar to those in dairy, so I'd avoid switching to that.
As I was already dairy free, it just meant feeding baby little bits of what I had.
For us weaning for the first few weeks was just offering tastes of food anyway. So things like strips of soft pear, banana etc.
There is no rush for them to be eating 'meals' and no need to be buying the off the shelf packaged baby foods unless you want to.
A dietician will be able to support you in safely following the milk ladder once baby is over 1yr.
They don't usually recommend trying to reintroduce dairy before then as they are unlikely to have grown out of it by then.