School-Based Respiratory Health Management Act This bill amends the Public Health Service Act to revise the conditions under which the Department of Health and Human Services, in making asthma-related grants, gives preference to a state. A preference is given to states that require elementary and secondary schools to have: (1) on the premises during operating hours at least one individual trained in the administration of rescue medications for allergies and reversible lower airway disorders, and (2) an allergies and reversible lower airway disorders management program. Currently, this preference is given to states that require schools to have on the premises an individual trained in the administration of epinephrine to a student having a severe allergic reaction. To be eligible for this preference, a state must have civil liability protections for trained individuals who administer rescue medications for allergies and reversible lower airway disorders.