Execute a Backup and Migrate backup via PHP
Ok, as it happens, I figured this one out. Below is a sample function meant to be used as an admin form submission handler. 

function _my_backup_handler() {
  $destination_hash = 'yourhash'; // See below.

  backup_migrate_include('profiles');
  $settings = _backup_migrate_profile_saved_default_profile();
  $settings-&gt;destination_id = $destination_hash;
  $settings-&gt;source_id = 'db'; // Your primary Drupal database. This is probably redundant.
  backup_migrate_perform_backup($settings);
}


As for what $destination_hash is, it turns out the "machine name" of a custom BM destination is a random hash. It can be found in the URL if you go to http://www.yoursite.com/admin/config/system/backup_migrate/destination and click the "edit" button for your destination; you'll end up at http://www.yoursite.com/admin/config/system/backup_migrate/destination/list/edit/[hash]. Just copy [hash] and plug it in as a string for the value of $settings-&gt;destination_id.