Thistle, I have heard of babies being allergic to breastmilk, and I also understand that it is very, very rare. I'd talk to several doctors and several lactation consultants, if I were you.
One thing: please don't listen to anything Dr. Sears says. He is a quack. Also, don't pay attention to random articles on the internet. I'm all for mothers breastfeeding their babies if they want to, and it does indeed seem like human milk (from a well-nourished mother) would be better for a baby than a substitute, but I also know that bonding takes place whether the milk comes from a breast or a bottle.
Also, regarding eczema: there has been a huge increase in allergies of all kinds in the past 20 years that does not appear to be related to the use of formula, since many more babies were formula-fed when I was an infant 40 years ago. So maybe formula triggers eczema and maybe it doesn't, but apparently it's not the main culprit.
I hope you figure it out! Do you mind me asking what makes you think your baby is allergic to breastmilk? My son got the runs all the time from breastmilk, but it wasn't because he was allergic to it. It was because he was only nursing for 10 minutes at a time, and therefore getting the early, more sugary milk only (and not the fatty milk produced as the nursing episode progresses). Or so they told me.