snangel, I'm so sorry you and your baby have so much to deal with. Sometimes life is just so unfair.
Your description of ecxema, mucousy poop, GERD and slow weight gain sounds like symptoms of food allergies. If any of those symptoms were present before starting solids I would say that the allergens are definately passing into your breastmilk. I think that is the case more often than not. So imo you should Do an elimination diet. It's really really hard, I think you should know that up front, but if you can do it, it's SO worth it when you see the allergic symptoms go away. One thing about an elimination diet is you will lose a bunch of weight really fast, most people find it disturbing, but it doesn't last. I dropped over 30 pounds in just a couple months but then things leveled out and I didn't lose any more. (darn). I also felt like crap and cried a lot, but that doesn't last either. I wish someone had told me about that stuff ahead of time so I wouldn't have worried about it so much. It's just your body adjusting. It will NOT affect your milk supply as long as you drink lots of water. And take your vitamins!
Obviously, I don't know the whole story here, but I find it REALLY weird that a doctor would recommend solids for weight gain. Ounce for ounce, breastmilk has more calories and useable nutrition than any solid food. And babies can only eat so much. I think that solids will make him cut down on nursing and may lead to even slower gain! Does he have trouble nursing or anything? Is he super skinny? At five months my daughter weighed 13 pounds also and she has food allergies. She is also just a petite person...she was never a fat baby but not bony or skinny looking either. Her doc thought her weight was fine even though she wasn't on the chart. I, of course, worried myself silly. (She is 2 now and has outgrown several allergies but is still allergic to eggs, milk, beef and pears. She weighs 23 pounds now).
What has your baby eaten that he has NOT shown an allergy to? If his gut is still immature giving him solids can CAUSE an allergy...my dd is allergic to pears and I think that's why. Pears are the least allergenic fruit there is.
You can do a modified elimination diet to start with and you might get lucky...that's where you just cut out the most common allergens and anything you suspect. Dairy, eggs, wheat, soy, nuts and seafood along with what you saw him react to. I did that and it didn't work, but I still think it's worth a try. The TED didn't work well for us either because she was allergic to rice. In hindsight it should have been obvious, but it wasn't until we did testing that we found it out, but I was suspecting by then. We did a RAST (blood test) at nine months and skin tests at 9 1/2 and 10 1/2 month. You need to do an elimination diet first though because they can only test babies for a limited number of things, so you need a list of "suspects".
Good luck with it, and remember whatever you choose to do dietwise, it's not forever and it WILL help your baby!