This is resolved. Thought I'd follow up with what was done in case anyone runs into the same problem.
In short, the TP communicates with the controller via Ethernet. When the TP locks as I described, it can be because the adapter is busy processing other traffic, and cannot dedicate cycles to communication with the TP.
I called tech support, and they told me that some users had experienced issues with Ethernet IP: In some cases, the controller network was connected to a high-traffic LAN, and the robot controller was looking at every packet whether it was directed at it or not - spinning its wheels processing these packets.
In my case, there is no other traffic on the LAN. However, I had configured and enabled the Ethernet IP port 1 in preparation for the PLC's interface, and I had port 2 configured to allow access from my PC to the controller's web page. When I disabled port 1, the problem corrected itself. Also, the PLC is now up and running, I have renabled Port 2 and am communicating with the PLC without issue, and am pulling up the controller's web page from port 1. In this configuration, I am no longer experiencing TP lock out.
Unfortunately, I can't give an absolute reason for the lockout. However, if your scenario is similar to mine, then I would suggest first disabling all Ethernet, and see if the issue resolves itself. If so, I would try communicating with the controller only through Port 2.
I hope that helps, and thanks to all who responded to my posts!