Not sure TMS supports AD LDS, not documented anywhere that I can find.
Do you have your TMS setup to use AD for user authentication under Admin Tools > Configuration > Network Settings > Active Directory?
Can both domains search for users within the other?
I'm not sure if TMS supports multiple domains or not, nothing saying it does/doesn't, however I found some old enhancements asking to support it: CSCub22236 and CSCup29892.
To log into TMS, users must have a Windows username and password that the server is configured to trust. By default, any local Windows user account will work, as well as any Active Directory domain user account if the server is a member of an Active Directory domain