My first question is this: How did WIC figure this out without a blood test ? (I didn't know WIC did blood tests ??)
Anyway, to find out the iron level in the blood, you have to do a hematocrit -- it's the blood test and it's simple, really. I would definately recommend that you get the blood drawn at the hospital, however, and not at the doctor's office. You want to be sure the people who do it are quick and good at it, so it doesn't hurt unnecessarily.
We went through this with our 2nd youngest, because the doctor ordered a routine test at 12 months. Her iron count was low. And the doctor said it's a concern because a lower than "normal" range iron level could make the child susceptible to diseases and infections.
Our daughter does not like meat either. She is 14 now, and eats a "no thank you" serving whenever we have meat. But she eats some things. She does eat meatloaf, and McD's burgers. (??) She prefers hot dogs with little nutritional value to hamburgers at home, however. She likes chicken.
So, try cheese. Does she like veggies? Green leafy veggies have iron in them. If she'll eat spinach leaves, then she can get iron there -- our food fusser also hates veggies. But she's still alive, and healthy. If the doctor's office orders the blood test, then, when you get the results, talk to the NURSE. Nurses have more training in nutrition and health than doctors do, because doctors spend so much time on diseases and abnormality. The nurse could give you information on building more iron into her diet. If you find a food she really likes, don't worry about feeding it to her more often than you think you should. Just get the iron in.
The other option becomes iron pills, and I gave my kids chewable multivitamins, but we had to work it until we found a brand that they kids liked. You can buy a straight multi-vitamin or one with iron. I'd try the regular one, and see if it's enough to bring the iron count up, and if not, then use the vitamins with iron.
Also be aware that you might need to add fruits when you add iron. Iron can cause constipation when taken in pill form.