I think there's a whole host of things probably going on here and I say this as a mum of two refluxers who both went on to have multiple intolerances and/or allergies.
Firstly, unless your DS has had a bug, lactose intolerance is unlikely. It is more likely to be a cows milk protein issue. (The protein, not the sugar in milk causing an issue.) Instead of buying a lactose-free milk, buy a dairy-free one.
Secondly, if you are still having difficulties with solid food by this age, I would be seeking a referral to a) a dietician and b) a specialist speech and language therapist who deals in sucking and swallowing/food aversion.
If you've got Coeliac disease in the family and ongoing gut and bowel issues in your DS, a simple blood test will help rule this out. Don't take any gluten out of the diet before having the test or the outcome will be affected.
Was your DS always sick after food? Was he refluxy as a baby? My youngest DS was sick 20 times a day until 14 months and I thought it would never end. He was medicated, but never very successfully. He was massive though, despite this! My eldest was a silent refluxer who lost a lot of weight and always struggled with milk. He is finally outgrowing his allergies during puberty, but most outgrow them earlier. (It's been a very long journey with both and we have Movicol here too for bowels.)
I would ask to see a different GP. (Maybe feed your DS in the waiting room so he is actually sick in front of the GP!!) They obviously aren't taking this seriously, but once kids develop food aversions, it can be really difficult to manage. Refuse to leave without the referrals I've mentioned. Take a partner or parent for support and cry if you have to. Maybe video it beforehand? (I got help when I finally took ds1 to the HV clinic in time for a feed. He screamed the place down for an hour while I battled to feed him and they witnessed how bad it was.)
If it is likely to be reflux, some anti-reflux meds will help in the first instance, but that in itself isn't enough here.
Good luck!