Leadership Recognition Act This bill makes the rate of monthly basic pay for the Senior Enlisted Advisors to the commanders of the combatant commands equivalent to the rate of monthly basic pay for the Senior Enlisted Advisor to the Chairman of the Joint Chiefs of Staff. The Department of Defense (DOD) shall submit to Congress a report setting forth an assessment of the advisability of including additional senior enlisted members among the most senior enlisted members who are entitled to an elevated rate of military basic pay. If DOD determines that the inclusion of any additional enlisted members or categories of enlisted members is advisable, the report shall specify such additions. The bill modifies the President's authority to determine alternative pay adjustments in the annual basic pay of members of the uniformed services to limit such adjustments to national emergency conditions.