Is Rash Like Symptoms Due To Milk Allergy In A 5 Month Old Child?

Good day, Doctor I would like to get some advice please. My baby is 5 months and vomits all the time. He does not look uncomfortable, he just burps then vomits. Sometimes the vomit comes without a burp. And does not seem forced. I have had him at his paed, who advised on a milk thickener for reflux. This did not work for him so I switched him to novalac ar plus losec which doc prescribed. Worked for a week and then back to vomiting. I put him back on nan which he started on, but made his poo dark green. And vomiting continued. Put him on to s26 gold... Again vomiting stopped for a week or 2 now it s back. He has also developed a rash around his mouth going down to his chin and on his cheeks. Looks flaky some days and other days not. The odd thing about the rash is that it is clear, but rough during the day.... But turns red at night? Could all of this be a milk allergy? What can I try? Thank you Marchalene