Hi A.,
her immune system is exhausted. like running marathons back to back with no rest.
sounds like she might be allergic to something or even emotionally afraid of something. perpetual fears even ones that are not 'big' to us can wear down immune systems.
Sometimes a different allergist can pick up something that the first one might have missed or was just not as apparent back then.
Also for the pneumonia and to restore her energy right now..."Black Elderberry" sugar free cough syrup is amazing.
sugar free is much better than that with sugar if you use the syrup.
i've had pneumonia so many times growing up and was otherwise a healthy child. since taking this black elderberry (go for a good quality brand), i haven't had it at all...for over 15 years! a brand name i use is 'Sambucol'. Wild Oast used to carry it and you can get it online. its kosher which means it is a very high quality and very consciously made. The Sambucol liquid syrup is a deep, deep crimson and is thinner than water so doing it over the sink is highly recommended!
another kind in many health food stores is made by Gaia Herbs and is in liquid capsule form...a six year old would probably want that over the liquid version...feels more like a vitamin.
the flu homeopathic flu remedy called "oscilloccoccinum" at most health stores (and at Walgreens now!) helps balance out the deep levels of one's system to rebuild. it is not a pharmaceutical and has no contraindications. it would probably take two rounds of it to make the deep shifts.
I don't know anything about igg's but i do know exhaustion from coughing...and these things helped tremendously. all can be used with young children as well. i've had many neighbors and friends use these with much success as well.
i see your daughter Happy, Healthy and Beaming with Sunshine and Laughter...deep breathing and enjoying all she does.
all the best!