I work in a hospital in the Medical Records department. Lately we have had many patients come back with similar situations to yours. I really feel your frustration. What I have found helpful and works in some cases is to actually call the doctors office ( I know, they are really hard to work with) or better yet show up in person and explain what happened. (who ever ordered the labs for you) tell them you were denied and that you more than likely would not have had the tests if you knew they wouldn't be covered. Most offices or hospitals know what is covered when you check in. The doctor should be aware or ask what your insurance is and know. There are times when a wrong code is put in. This code can usually only be changed by asking the coding people in writing, coming from the doctor. If it is an error made by the coding department (and you probably won't know without discussing this with the doctors office or hospital, in which case it is a little different)
It can be fixed in some instances. This may be different in your state I really do not know, but that is what I have seen done here.
Good luck