Thank you for sharing your situation with the Wefugee Community!
Unfortunately, I am not familiar with (private) Universities' systems and their fees regulations.
That policy of exempting students from Syria, Iraq, Somalia and Eritrea (maybe Iran also?!) from paying the fees probably derives from the outdated (since 1.8.2019) approach and assumption, that asylum seekers from those countries have a higher chance of getting a positive decision from the BAMF and international protection accordingly. Officially, their quota of receiving that status was more than 50%. A few weeks ago, that policy was revised and now only Syrian and Eritrean nationals are still privileged in that sense.
Maybe you can try to convince the admin staff or people in charge (in case also asking for their supervisors) with an argumentation referring to the current official protection data and statistics. As you can see i.e. in this analysis published by the BAMF (see page 3 esp.), the protection quota for Turkish nationals is nowadays even higher than the one for applicants from Somali, Iran or Iraq.
Maybe a lawyer could also support you in drafting a paper or letter to the University disagreeing with their decision and describing your situation and bringing forward your arguments.
I wish you all the best and please do not hesitate to get back to us, if you want to discuss topics further.