It sounds like your son might be allergic to something in your wife's diet. Both dairy and gluten can be common causes. Are there any known diet issues in your families to give you a place to start? My son had dairy allergies, and would react to *any* dairy in my diet (so I got good at reading labels). He was much better after I made that change, though in his case we also needed reflux medicine as he was very bad. So definitely something to try if you'd like to continue breastfeeding.
However, modern formula milk is very good, so if you decide to use that rather than breastfeed it really is up to you. Your son will grow up healthy and loved regardless :)