Can you set a document attribute on an outbound document through TN?
Yes, you can.
Create the processing rule to call a flow service to set it.
There is a WmTN built-in service available (wm.tn.doc:setAttribute).
Another option is When you creating a TN DocType, there itself you can create a custom document attribute and extract the value from the document which is routed to TN.
HTH,