Foreign Agents Disclosure and Registration Enhancement Act of 2021 This bill provides the Department of Justice (DOJ) with a mechanism to demand evidence for investigating compliance with foreign-agent disclosure requirements and increases penalties relating to such requirements. DOJ may issue civil investigative demands to a person who may have information relevant to an investigation related to the Foreign Agents Registration Act of 1938 (FARA). Such demands may be for documents, written answers to questions, or oral testimony. The bill prescribes various procedures and requirements, such as minimum notice periods when seeking oral testimony. Investigative demands may be enforced or challenged in district court. Such investigative demands may not require the production of information that would be protected from disclosure under (1) the rules governing grand jury subpoenas, or (2) the Federal Rules of Civil Procedure to the extent the rules are consistent with this bill. The bill increases the maximum criminal fine for certain FARA violations from $10,000 to $200,000. It shall be unlawful for an agent of a foreign principal to willfully fail to disclose being a FARA-registered agent before or during a meeting with a Member of Congress (or with staff of a Member or congressional committee). The bill provides for various civil penalties for failing to meet agent registration requirements. The foreign principal of a penalized agent may not pay the imposed fines. The Government Accountability Office shall (1) analyze the effectiveness of FARA enforcement, and (2) audit the exemption to the Lobbying Disclosure Act for FARA-registered agents.