Thanks for the tip. I added the RestartThing to my code just in case. It seems I have to close and re-open the thing in the composer to see the bindings. Refreshing the view does not update the binding status—only the values.
When I open the "Manage Bindings" I can now see the bound properties. There are also the remove binding button on each property except the problematic one. I still cannot unbind it either programmatically or manually. I restarted Tomcat but it did not help. Probably something weird in the TW platform. I'll probably need to delete the thing and re-create it.