Success!
The issue I had earlier was that I incorrectly ordered the dual channel version of the LVDS-to-HDMI converter. This is the default version they ship to you, and it is designed for 1920x1080 output. You have to specifically ask to order the single channel (“Single-8”) version that will properly convert the 1366x768 output. I was able to capture the output with an Elgato Cam Link without issue. The splitter also splits the video so you maintain the normal display on the actual game itself. You can snake the HDMI cable out of the game, close it up, and capture that perfect stream quality. 
Happy to share this with everyone out there, it was a lot of work, but perhaps it’s helpful to someone else.