Dear Nate
I cannot help that much. I'm also trying to connect the CW paddle via the RS232 port to the rig as "explained" in the manual, but its does not work straight away. It seems that some software is needed, which is not yet available.
So far I got a few steps further being able to connect a vertical key using the RadioSync software, which comes with the EESDR3.
I would suggest that you try to communicate via TCI with Radio Sync, suing the TCI configuration and then let RadioSync communicate with EESDR3, which should happen easily, showing up with green lights.
The good news is that the configuration options in RadioSync are limited and you can try them "all", if you can't find any help for doing so.
I hope this helps, at leas as a way marker what you can try next.
73 de EA5Q
Stefan