Oh mama, firstly hope you're feeling better from the stomach flu, that must have been such a tough time for the whole family! The rash around the mouth appearing with every formula brand you've tried does sound like it could be Cow's Milk Protein Allergy (CMPA), where baby reacts specifically to the cow's milk protein in the formula, and since Lactogen, Kendamil, and Karihome all contain cow's milk, that pattern makes sense. A friend of mine went through something very similar where her baby would get localised redness right where the formula touched the skin, and it turned out to be exactly that. Since the rash is consistently appearing across multiple brands and your baby is only six months old and still quite young, I'd really encourage you to loop back with your doctor or a paediatric allergist before continuing the formula, so they can properly assess whether it's CMPA and possibly recommend a hypoallergenic or extensively hydrolysed formula that's gentler on sensitive tummies!