Hi Laura,
Yes, it sounds exactly like Milk Protein Intolerance
My DS has this, and we got a prescription formula called Aptamil Pepti 1 to give him. Since he was on that his skin cleared up and his digestive problems went and he finally started putting on weight
Reflux doesn't cause eczema, MPI does. Go back to your GP and ask for a trial of Pepti 1.
If she's still suffering after 2-4 weeks on Pepti, she needs an elemental formula such as Neocate which contains no milk protein at all. (Pepti does but the proteins are broken down so baby's gut doesn't recognise them as allergens).
MPI is not at all uncommon, many of us here have experience of having a child with it. DS also had reflux (associated with his MPI) but Gaviscon didn't help, it just constipated him. Instead we dosed him with Ranitidine until he outgrew his reflux at 13mo once he was taking more solids.
Good luck with the GP, and do keep pushing for proper help because gut damage will cause your LO unneccessary pain and there is a standard course of action to follow to sort it out (as explained above).