If nobody comes up with a parameter to override the
privateweb restriction you could hack the code.
There are many reasons not to do this but if you have to, look in assets/snippets/wayfinder/wyfinder.inc.php around line 349.
Change this =>
// build query
$access = ($modx->isFrontend() ? "sc.privateweb=0" : "1='{$_SESSION['mgrRole']}' OR sc.privatemgr=0").(!$docgrp ? "" : " OR dg.document_group IN ({$docgrp})");
$sql = "SELECT DISTINCT {$fields} FROM {$tblsc} sc LEFT JOIN {$tbldg} dg ON dg.document = sc.id WHERE sc.published=1 AND sc.deleted=0 AND ({$access}){$menuWhere} AND sc.id IN (".implode(',',$ids).") GROUP BY sc.id ORDER BY {$sort} {$this->_config['sortOrder']} {$sqlLimit};";
To this =>
// build query
$sql = "SELECT DISTINCT {$fields} FROM {$tblsc} sc LEFT JOIN {$tbldg} dg ON dg.document = sc.id WHERE sc.published=1 AND sc.deleted=0 {$menuWhere} AND sc.id IN (".implode(',',$ids).") GROUP BY sc.id ORDER BY {$sort} {$this->_config['sortOrder']} {$sqlLimit};";
It removes the check for whether the document is private.
The private document appears in the menu but when clicked the user is redirected to the unauthorized page.
Make sure your unauthorised page in the site configuration is set to your register page.
I’ve only tested this quickly on a single level menu.
Backup the original file in case it goes belly up and you need to revert.
Test it locally first if you can.
Good luck