The best course of action for the FI in this scenario is to consider retaining qualified, experienced legal counsel. This is because the investigation request from the local law enforcement agency may involve complex legal issues, such as confidentiality, privacy, data protection, and cooperation obligations. The FI may need to balance the interests of the customer, the law enforcement agency, and the FI itself, and ensure that it complies with the applicable laws and regulations in responding to the request. The legal counsel can advise the FI on the scope, validity, and implications of the request, and help the FI to prepare and submit the required information in a timely and appropriate manner. The legal counsel can also protect the FI from any potential liability or litigation arising from the investigation.
References:
1: Requests by Law Enforcement for Financial Institutions to Maintain Accounts and/or Conduct Transactions1
2: Answers to Frequently Asked Questions Regarding Suspicious Activity Reporting and Other Anti-Money Laundering Considerations2
3: ACAMS Study Guide for the CAMS Certification Examination3

The most important consideration when deciding whether to recommend closing the account is the requests from the competent authority, such as the Financial Intelligence Unit (FIU), the regulator, or the law enforcement. According to the FAQs related to Suspicious Transaction Reporting issued by the Financial Monitoring Unit of Pakistan1, reporting entities should not terminate the relationship with the customer after filing a STR, unless instructed by the competent authority. This is because closing the account may alert the customer of the STR, compromise the investigation, or hinder the collection of further evidence. Therefore, the reporting entity should consult with the competent authority before taking any action to close the account.
The other options are less important or irrelevant considerations. The institution's anti-money laundering policy may provide some guidance on how to handle high-risk customers or accounts, but it should not override the requests from the competent authority. Customer relations and the Chief Executive's reputational risk are not valid reasons to keep the account open if there is evidence of money laundering or terrorist financing. The reporting entity should act in accordance with the law and the best interests of the public, not the personal or business interests of the customer or the bank's management.
References:
1: Frequently Asked Questions (FAQs) related to Suspicious Transaction Reporting, 5

"The first European Union Directive, "Prevention of the Use of the Financial System for the Purpose of Money Laundering (Directive 91/308/EEC)," was adopted by the Council of the European Communities in June 1991. Like all directives adopted by the Council, it required member states to achieve (by amending national law, if necessary) the specified results. This First Directive required the members to enact legislation to prevent their domestic financial systems from being used for money laundering"