There are websites with photos of various rashes, bites, skin problems, which might help you in narrowing down what your problem is. You can google things like "photos of rashes" or "identify rash" and you will see a low of help there. And there are many websites that help in identifying bed bug bites and clues. Take a photo of the rash when it is burning so that you have good information to bring to a doctor.
I googled this, and believe it or not, there are some women who develop hives and allergic reactions from breastfeeding! There are entire forums devoted to the subject. Apparently, the culprit is certain hormones or chemicals (I'm no scientist) that are released in order to breastfeed, and some women are allergic! You may want to consult an allergist. Try googling "hives from breastfeeding". Many women were told they were crazy and it was impossible, but you may find their stories interesting, and the relief they found when they learned that it is possible is overwhelming!