Uggggh - insurance issues are the worst!!!!!!
Not sure this well help, but having any Dr is better than NO Dr, so I thought I would throw it out there in case you are stuck in a pinch and really need a Dr. Also, they take KidCare.
However, I don't think she has access to Children's memorial and she is NOT a pediatrician (I didn't use a pediatrician for my daughter, we have a general practitioner) and they don't have pediatricians at their office, but I have gone to Dr Tanya Richardson with Ressurection. She is at 711 W North Ave (in the North/Clyborn/Halstead area).
Both my daugther and I have seen Dr Richardson and Dr James Nee and I like them both. They are very thorough and will answer any of my questions as well as having a good rapport with my daughter (she is older now - but we have been going there since she was 5).
Incidentally, Dr Nee diagnosed a Staph infection in ME after I had been misdiagnosed at 3 other places (2 Drs while on vacation and then the ER) so I really trust the 2 of them. I can't really speak for any of the other Dr's in their practice, I only go see one of the 2 of them.
Good Luck!