Extend the access input's view:

$access_id = elgg_extract('access_id', $vars, ACCESS_DEFAULT);

if (!elgg_is_admin_logged_in()) {
    $access_options[ACCESS_LOGGED_IN] = elgg_echo('LOGGED_IN');
   
    $user_guid = elgg_get_logged_in_user_guid();
   
    $dbprefix = elgg_get_config('dbprefix');

    $query = "SELECT * FROM {$dbprefix}access_collections
             WHERE owner_guid = {$user_guid}
             AND site_guid = 1
             ORDER BY name ASC";
    $collections = get_data_row($query);

    foreach($collections as $collection){
        $access_options[$collection->id] = $collection->name;
    }
   
    $field = [
        '#type' => 'access',
        'name' => 'access_id',
        'value' => $access_id,
        'options_values' => $access_options,
        'entity_type' => 'object',
        'entity_subtype' => 'hjwall',
        '#label' => elgg_echo('access'),
    ];

    echo elgg_view_field($field);
}

Big thanks!