The question is irrelevant. The church was built by Christians and stolen from them by force of arms. If Turkey would offer it to the Ecumenical Patriarch I am sure that he would come up with the funds to upkeep it. Such a move would cause no damage to the Muslims since it is no longer used as a mosque.
Islam claims that it is a religion of peace and diversity. What better way to prove it? If Turkey truly wants to join the Western nations in the EU then they must be held to the same standards. Christians in Turkey should have the same rights and freedoms as Muslims in the EU.