I have so many friends who went through this. One had a 1 year old who had a whole winter of 3 weeks sick/one week well. They did everything you are doing - mattress incline, agonizing over animals, and considering the testing. He had pneumonia-like symptoms and used a nebulizer constantly, but they worried about the chemicals/drugs he was inhaling every time. The mom's allergist took one look at the kid and said, "He's going to be MINE for years." When someone gets the allergy "prick test" and is positive for 76 of the 80 things like one friend was, you know there is a lot going on. The mom was on biweekly shots and 2 prescription meds, and was still miserable. She did not want to put her child through this.
You (and your doctor) may not want to put a 2 year old through the testing at this point. From your description, it appears that your child either has infections or allergies - so there is either an immune system that needs boosting, or an imbalance that is causing an allergic response. Or a combination of both. From my professional experience, I am well aware that "eating well" is really almost impossible nowadays because our foods are either over-processed or grown in weakened soils with insufficient nutrients due to over-farming. Although most doctors take a "medication" approach because that is how they have been trained and because most of their sales visits are from pharmaceutical salespeople, the AMA long ago said that people need to supplement. However, most supplements are in pill form or are not really scientifically formulated for maximum absorption and proper balance. SO people generally waste their money and don't have particularly good results.
If you are interested in a nutritional supplement designed for and absolutely safe for children (even infants) and safe as a pre-natal even, I can help guide you to a happier daughter and a happier you. This is what my friend with the allergies did - found another way. Another friend has 3 daughters who had terrible allergies and asthma - now gone. As they have helped me, I am spreading the word to others. Doing something nutritionally with patented products (therefore unique, safe and effective) is a phenomenal alternative to drugs and endless testing. Happy to help!