Is it permissible to file a lawsuit against the Passport Department to obtain a passport to perform the Hajj pilgrimage with my mother, given their obstacles and the delay in issuing the passport, and would this be considered disobedience to the ruler?

Filing a lawsuit to obtain one's right and remove injustice is permissible in Islamic law. However, the benefits and harms resulting from it must be assessed, for Islamic law is built upon achieving benefits and averting harms. If benefits and harms conflict, then achieving the greater of the two benefits or averting the greater of the two harms is given precedence. Moreover, suing various entities is not considered disobedience to the ruler (wali al-amr), because it is they who have established the judicial system.