I totally understand your frustration, and it sounds like he definitely is allergic to something(s). I struggled with breastfeeding and allergies with my son. He had multiple food allergies- some serious- and I don't think I would have ever figured out the exact combination on my own. He had horrible colic- screaming in pain, spit up bad after breastfeeding, green frothy stools- it was terrible. What did I do? I gave up breastfeeding when he was 4 1/2 months. Seriously, I just couldn't take the screaming any more. As soon as I gave him formula, the colic subsided (he is NOT allergic to diary). I always said if I had of known that it would fix the colic and other digestive problems, I would have introduced formula a long time before! There are just some cases were breastfeeding is not best, and in my case was one of them. I subjected him to months of torture when I didn't have to- I felt guilty for forcing the breastfeeding on him after we finally had him tested. :( Before I get hate mail, I am not discouraging you from breastfeeding in any way. I am just giving you my personal experience.
When my son gets those red circles under his eyes, then he has been exposed to something he's allergic to. They are called allergic shiners. With his sensitive skin, I gave plain baths- no soaps except for California Baby super sensitive skin wash when I needed to wash his hair. Also, no perfumes in any of the detergents, including dryer sheets.
My pediatrician finally recommended an allergist when he was 11 mos old and had a very scary reaction to some egg. We found, among other things, he had a potentially life threatening allergy to eggs. I am not sure about your allergist, but mine doesn't typically take babies younger than 6 mos, so may have to wait. Just don't do any high allergy foods, and introduce one food at a time slowly over the course of a couple of days.
Also, I always hand wash my bottles. The dishwasher always leaves behind some residue. If you do wash in the dishwasher, make sure to rinse them again thoroughly before giving to baby. I also sanitize as an added precaution.
Hope this helps!