In addition to adding the domain to the Management Group, you will also need to add Credentials to the Credential Rings, so that the Patch Manager console user can enumerate the domain. Most likely the domain is not appearing in the Windows Networking node because there is no authority to query the Domain Controller.
Now... aside from that, there WILL be issues managing two domains that are using the same NETBIOS domain name. At a minimum this will likely require you to always specify domain identies using username@domain.local format rather than DOMAIN\username format, since it will be impossible to distinguish DOMAIN1 from DOMAIN2 using the DOMAIN\username format.