Dequeue script in template isn’t working

Move your project_dequeue_unnecessary_scripts() function to your functions.php file and add a conditional statement to determine if the appropriate template is being loaded. E.g.:

// Remove Mobile Header
function project_dequeue_unnecessary_scripts() {
    if ( is_page_template( 'name-of-template.php' ) ) {
        wp_dequeue_script( 'enterprise-responsive-menu' );
        wp_deregister_script( 'enterprise-responsive-menu' );
    }
}
add_action( 'wp_print_scripts', 'project_dequeue_unnecessary_scripts' );

I suspect that your function is not working because it has been placed somewhere after the call to get_header() in the template file which means it would be too late to dequeue the script. Declaring functions in template files is not a good practice anyway, so use your functions.php file or another include.