Accessing Active Directory LDAP with SSL connection over the Internet

Here's the situation:  We have a vendor, hosted outside of our local network, that we want people at our company to be able to authenticate to using our internal LDAP.  We don't have an Active Directory, or other LDAP server in our DMZ capable of connecting to their servers.  All of our AD servers reside in our internal network.

Here's what we did:  we made it so one of our Active Directory servers is accessible from the outside on port 636, but ONLY to the public IP addresses being used by the vendor.  That part in itself is working fine (a telnet connection to the public IP on port 636 is successful).  The server (names changed for this example obviously) is a member of a local AD domain, and the name on the network is "server1.inside.local" (internal.local is our internal domain).  Obviously the vendor won't be able to resolve to that.  So, we bought a certificate for "server1.outside.edu" (our outside domain), applied it to the server, and added some DNS entries so our network knows to send requests for both "server1.inside.local" as well as "server1.outside.edu" to this same box.  

The problem we're having:  the LDAP communications need to be secure.  We bought a certificate for "server1.outside.edu" and applied it to the server.  As far as I can tell, the cert is installed correctly, as is the intermediate CA for the cert.  However, using Microsoft's ldp.exe utility, I can't connect to the server on port 636 at "server1.outside.edu."  I can only connect to the internal DNS name, "server1.internal.local."  Even locally&for now, forget that ultimately we want an outside vendor to connect.  I can't even connect with ldp.exe using the public name when I'm on a computer that's inside of our network -- nothing in between the server and the desktop but a switch, no firewall.  I've spent hours on this, tried everything I could think of.  I went through these instructions from Microsoft..no luck:  http://support.microsoft.com/kb/321051   I'm not sure what else to do.  How do I make this work, or should I be approaching this in another way?

So to sum up:
Server's local name:  server1.internal.local
Server's outside name (applied by DNS entry):  server1.outside.edu
Certificate for server1.outside.edu is applied to the server, but can't make a secure LDAP connection to the server using that name.  

Hopefully I explained this clearly but if anything doesn't make sense, let me know.