Hi @fieldset and thanks for your query. I am afraid that the Tobii Gaze Trace functionality does not offer the feature to specify a particular monitor to stream to, but perhaps you have already looked at Tobii Ghost? This would allow you to stream the gaze overlay to a different computer in turn connected to the TV which may be a workaround?
You can also look at using the Tobii Streaming Gaze Overlay which offer similar functionality and you can read more about @ https://blog.tobii.com/introducing-streaming-gaze-overlay-d4b02c87430b
In terms of using only one system, You could conceivably set up a virtual machine and run this as a virtual second PC to connect to the host via Ghost or TSGO although I am not sure such a procedure has been tested before!
Please let us know if you think any of these solutions are viable and if we could be of any further assistance.
Update: Tobii Ghost is a new name for Tobii Streaming Gaze Overlay and you don’t need any virtual machines for such setup.