As a parent of a child with allergies to wheat, dairy, soy, eggs, potato, peanut and grapefruit, I have this to say...
First off, Eczema *IS* a sign of an allergy or intolerance (not always--but it is a huge sign)...It was my sons first sign, and a way that I know he had an infraction...
Second, not showing outward signs of an allergy tells you very little, since most allergies effect the inside of your body--food wise, this can cause tears in the lining of the stomach, asthma can develop, lung issues, brain function and other thing that you do not see. Depending on how high her levels are for allergies to cats, depends on if the cat needs a new home. ME? I wouldnt take the chance that this cat is making her sick, or causing internal damage. The way allergies work is this--the more you are exposed to an allergen, the worse the allergy will get. If you keep her away from her allergens, this may lessen over time, and eventually, she may be able to tolerate some of her allergens.
Dont be fooled by doctors telling you that they can not determine an allergy till they are 3 years old...This is a median age that most allergies will develop by, this does not mean that allergies can not be exist and be corrected ahead of time, to avoid health problems and damage done. The earlier its caught, the better! I wish I had listened and did more research when my son was 4 months old, vomiting so much that we were afraid we were going to lose him...and we were told by the #1 ped in town that we could not test for allergies till he was 3 years old--It was a load of bull...because we didnt test till he was about 2 1/2, and a different ped/state later, we have some major health issues with our son.
I noticed you said you were avoiding dairy, and took that as you believe eggs are dairy...this is a common mistake. Eggs are not dairy. There are also MANY egg replacers and a lot of different foods that are made without eggs, so life can be hard at first discovering all of it, but can get easier as you learn.
Personally, I feel you need to avoid eggs, peanuts and most nuts in general until you know what her levels are, because its the proteins that are being transferred with your breast milk. These can make her allergy worse, and you dont want to be at the point of having to carry an epi pen with you every where you go.
And the studies of exposure to pets--this was showing that if you have pets growing up, you are less likely to develop allergies---but if you already have the allergy...the more you are exposed, the worse the allergy becomes.
These are my opinions based on my personal experience growing up with allergies, and now have two kids with numerous allergies.
I think its great that you caught this so early and that you are taking the right steps to help her out! :) Great for you! You are a great Mom!