The trick here is when to remove it as this is set based on the value of:
$do_filter_list

That changes based on:

1 go through via manufactuer and get filter on Categories

2 go through via category and get filter on Manufactuerers

This is setup in the file:
/includes/index_filters/default_filter.php

There isn't an override for this file nor a switch for that dropdown ...

The code around line 166 controls this:
PHP Code:
    $do_filter_list false;
    
$filterlist $db->Execute($filterlist_sql);
    if (
$filterlist->RecordCount() > 1) {
        
$do_filter_list true;
      if (isset(
$_GET['manufacturers_id'])) {
        
$getoption_set =  true;
        
$get_option_variable 'manufacturers_id';
        
$options = array(array('id' => '''text' => TEXT_ALL_CATEGORIES));
      } else {
        
$options = array(array('id' => '''text' => TEXT_ALL_MANUFACTURERS));
      }
      while (!
$filterlist->EOF) {
        
$options[] = array('id' => $filterlist->fields['id'], 'text' => $filterlist->fields['name']);
        
$filterlist->MoveNext();
      }
    }