Federal Student Loan Integrity Act This bill limits the authority of the Department of Education (ED) to waive or modify statutes and regulations in response to military contingencies or national emergencies, including by prohibiting ED from further using this authority in connection with the COVID-19 national emergency. Under the Higher Education Relief Opportunities for Students (HEROES) Act of 2003, ED may waive or modify any statutory or regulatory provision applicable to federal student-aid programs as ED deems necessary in connection with a war or other military operation or national emergency. This bill prohibits ED from issuing a waiver or modification that (1) provides for a period that exceeds 30 days during which payments of principal or interest due on federal student loans are suspended or interest does not accrue on such loans, or (2) results in the discharge or cancellation of federal student loans. Further, the bill prohibits ED from using this authority to waive or modify any statutory or regulatory provision applicable to federal student-aid programs in connection with the national emergency declared by the President on March 13, 2020, to respond to COVID-19.