Keep in mind that when members agree to display their information in the directory, they are essentially agreeing to have their information visible to their stake members (See the "What is this?" link in your household's profile). They are not agreeing to having their information available to members outside their stake.
Out-of-unit ecclesiastical leaders are given access to the units they are called to serve in BECAUSE they are given ecclesiastical authority for that unit.
Perhaps members of your stake would agree to give their information to the Mission President, but the Directory application is designed to follow the correct lines of authority when granting access, so be careful and thoughtful when circumventing the design of the application.