Have had a bit of a look about lactose intolerance and it seems that it is pretty rare in babies and tends to cause GI symptoms - not so much eczema. Of course this is just taking a look at google - not medical expertise!
If removing dairy from a diet it is important to replace calcium, fats and proteins. We have been given a calcium supplement for Olly but you would be better of maybe adding an extra portion of a calcium rich food. There are lots of examples of these around
. For protein, add a bit more meat/soy/lentil etc and fats some coconut milk or avacado etc.
How was your DS when he was younger and on a milk only diet? Did you notice any symptoms then?