Financial Stability Oversight Council Insurance Member Continuity Act This bill amends the Financial Stability Act of 2010 to modify the term of the independent member with insurance expertise of the Financial Stability Oversight Council to allow the current member to continue to serve in the event of a delay in the naming of a successor by the end of the member's six-year term. In such a case, the independent member may continue to serve until 18 months after the end of the term or until a successor is confirmed, whichever is earlier.