F.M.
Does the zinc levels cause her to be sick often? I would think so. Can you try the lozenges and the supplement? I am not sure. I think seafood has zinc in it, but if she is a slow eater, I don't know if you could even get her try it, unless she enjoys the peeling the shrimp, then she may eat more. Find out what zinc foods would give her an interesting way to eat. Like giving a toothpick to small children to eat their beans. They LOVE to do that.
Also, my son has been sick often too, and I was very heppy with our clinic. When we started to check for a Ped because of his special issues (and then our FP became a pain), we found that the ped office could actually get sick children in the same day, rather than ending up in the ER. I never even considered that any clinic would do that, ours could only sometimes. Some children can wait, and others can't, when trying to prevent inpatient stays. So, you may look into getting with a doctor that can see their sick children in that day.