Peanut butter, coconut oil...
No idea how bad her eczema is, but I do know that allergy testing (scratch tests I mean) are really hard to do for infants. Both painfully/emotionally hard and also tons-of-false-negatives hard.
Maybe they are taking blood to test specifically and maybe that would be more accurate. I would think a pb & coconut oil & frozen fruit smoothly a few times a day could help the poundage and sounds delish to me :-) Good luck
Avocados, nuts, lamb (organic if you can get it, but it's really fatty).
We have a ton of this in my family, neither I nor my kids can eat many forms of dairy. My sister has the same problem as you (with allergies and Mama tininess), but she really just doesn't like to eat fatty foods and will be going to formula for her FTT fourth month old. This is her second, and for whatever reason she just can't eat fatty food in enough quantity to make milk. For the record my second child was also FTT (failure to thrive) from 3-9 months due to undiagnosed allergies and we did the scratch tests.
I get the "only organic" idea, but right now you are trying to figure out stuff you can gain/maintain your weight on and make milk so baby does the same. Figure out what food sources work first, and then get the organic varieties when possible!