Ugh, how frustrating. I didn't realize that happened in Chicago although I've heard about it in other states - crazy.
I don't have experience with this, but from this policy it looks to me like they are required to evaluate for grade placement and can't just make a decision based on age (page 4) http://policy.cps.k12.il.us/documents/702.1.pdf
1. Grade Placement. Students who were previously enrolled in a private school orother school district who seek enrollment in the Chicago Public Schools in any grade may be evaluated for appropriate grade placement. Further, schools shall
follow the Office of Specialized Services’ Procedural Manual on students with disabilities transferring from other school districts.
Also, in the paragraph where it talks about age, it uses the word "May," not "Must." ("In accordance with Board Rule 6-2, children may enroll in kindergarten or 1st grade if they have reached the age of 5 or 6 years respectively on or before September 1st.") I'm not an expert, but it sounds to me like the school officials are interpreting the policy the way they want to.
Keep doing what you're doing and asking other parents for advice. If there is some sort of special ed advocacy group working with parents in Chicago, that might be a good starting point. Good luck!