I read just about all of the articles I could find on troubleshooting the double-hop issue, and I finally solved the problem thanks to some help from a really smart co-worker.
- SQL Server Machine and App Server were both trusted for delegation
- SQL Server was running as a domain account that was also trusted for delegation
- There was an SPN for the domain account referencing the SQL Server machine
The kicker ended up being the way that the domain was set up, it would not accept the short name for the server, I had to use the FQDN instead (as that is how the SPN was set up....usually I don't have this problem though).
Once I changed the SQLCMD -S parameter to use the FQDN (servername.subdomain.domain.local) it worked like a charm. The big pain in the ass was that the Kerberos Config manager was giving me no problems when I checked it using the short name, so I was going down a lot of rabbit holes.
In any case, it is resolved now and thanks so much for responding here!