I had a similar problem with my two children, down to the sports induced asthma, though the source of allergies in my son were dust, pollen and other foods. Both of my children had atopic eczema. My son had sensibility to cow's milk, egg, fish, some fruits -like lemons or strawberries- and some vegetables -like tomatoes- when he was little.
It looks like you are doing well. I don't think you are feeding her too much. You don't need to worry about how much she eats as long as her weight is between the normal range. Babies grow at different rates.
I would start adding some meat to the midday meal. Around six months old iron deposits laid down before birth are almost depleted. The recommendation from my pediatrician was to use red meat; apparently it had less hormones than chicken. As you only need 1-2 oz per day until they are over 12 months, I choose the leanest organic beef I could find.
I prepared my own food - carrot, peeled and diced, potato, peeled and diced, a few green beans, washed and trimmed, meat and simmer it until it was tender, and drizzling a little olive oil at the end. Then liquidize it -a hand held blender is great for this task.
You can prepare enough for a few days and freeze it in ice trays, then transfer the cubes to a bag and defrost what you need. Homemade instant food.
My children's eczema would flare when they had enriched food or vitamin drops, so I had to go the organic way for everything, which is expensive. They are still allergic to pollen and dust -though they are much better, no asthma and barely medication- but most of their food sensibility has disappeared. Unhappily, nut allergies do not tend to disappear with age.
I wish you the best.