At this point I believe it is a TCP port being blocked by the 1783-NATR.
I wanted to share that even though your camera is using EthernetIP to communicate with the controller, on the Dataman setup tool side, you don't use EthernetIP. Non EthernetIP TCP ports are sometimes blocked by Rockwell NATs, an example to unlock them is here:
http://www.plctalk.net/qanda/showthread.php?p=896370#post896370
Finding the blocked port number:
In the dataman setup tool manual:
https://support.cognex.com/docs/dmst_616SR1/EN/SetupTool_Reference_Manual.pdf
Page 43 mentions Telnet ports needing to be open. Using a combination of wireshark and Telnet, you should be able to see which ports are blocked by the NATR, and then add them to be unblocked in the "Custom rules" button on your NATR's translation table.
You can also call Cognex to request a list of ports they use for their software to work.