Is there any way to connect to one server and utilize a linked server using Windows authentication and NTLM?
Without setting up Kerberos then no.  Why do they not want to setup Kerberos?  It's already 99% setup.  All you need to do is create the SPNs and restart the SQL Server instances.  Kerberos is the default AD authentication system.  You are falling back to the less secure authentication systems.  Sounds like the powers that be don't understand the differences.