Sounds very similar to DS's symptons, he also had mild excma, he has ben fine since we have taken him of any cow dairy products, he seems to tolerate goats milk in cereal, but to young to try for a full drink yet, and also soya milk, but anything with milk makes hime throw up and dodgy nappies and very uncomfortable.
I would only swap on the advice of a dietician. Soya/goats doesn't have the correct sort of fat and even with the added calcium is still deficient compared to cow's milk. If he is growing, gaining weight his skin is clear and he is in no discomfort I would not swap.
well for formula the boys all had soa formula; ds3 had enfamil lactofree whixch wasnt exactly right for him (being casein intol) but was a heck of a good milk anyway (tasted locely, I used to sue it also as les reactive on me than proper milk)
DS4 is still ebing BF with me following a caseiin free diet, however he will have bits of soya formula in his meals if needed.
AS they are older the boys have moved onto goats milk, and seem fine on it. The only issue is that oyou cant get it in small cartons for school (to replace the standard free milk) so ds3 has Rice Cream which you can buy in Sainsburys
In 'the old days' when we had ds1 the soya available had terribly high sugars and ds1 lost most of his teeth but they seem to have resolved that now which makes it less baby nice tqasting but better if you can them to drink it I think.
After 3 ordinary rice / soya is fine but before that they need more fat.