Is it possible to filter the main loop to exclude posts from a specific category?

You still are going to make use of pre_get_posts to alter the main query query vars before the main query actually runs. The blog page/homepage can be targeted with the is_home() conditional tag

add_action( 'pre_get_posts', function ( $q )
{
    if (    is_home() // Only targets the home/blog page
         && $q->is_main_query() // Only targets the main query
    ) {
        // Works on build in taxonomy category, for custom taxonomies, use a tax_query
        $q->set( 'cat', -1 ); // Change with correct category ID, minus sign means remove
    }
}, PHP_MAX_INT ):

I would add this into a plugin which you can activate (as Super Admin) as a network plugin across the entire network