I've worked with a few customers recently that have taken Duet Enterprise lists offline to Office 2010 clients. In order to troubleshoot issues related to using Business Connectivity Services offline you really need to use the BCS client side logging. The following TechNet article has a nice description of how to enable Business Connectivity Services logging on the client machine.
Use tracing on the client (SharePoint Server 2010)
Here are the condensed steps that I use on the client when troubleshooting issues related to taking Duet Enterprise lists offline.
NOTE: You must be an Administrator on the computer to use the tracing.
Create the Data Collector Set:
1. Launch Perfmon (Start --> Run --> Perfmon)
2. Expand Data Collector Sets
3. Right-click on "User Defined" and choose "New Data Collector set"
4. Give it a name, I use "BCS"
5. Choose the "Create Manually" option and click Next
6. Check the box labeled "Event trace data" and click Next
7. Next to the Providers box, click the "Add..." button and wait for the list to load.
8. Select the item named "Microsoft-Office-Business Connectivity Services" and click "OK"
9. Leave everything at the default values and click "Finish"
10. You should now see your "BCS" Data Collector Set listed under "User Defined" in Perfmon.
Start collecting the trace information:
1. Select the "BCS" data collector set you created previously.
2. Click the "Start the Data Collector Set" button.
3. Reproduce the issue.
4. Click the "Stop the Data Collector Set" button to stop the trace.
5. A trace file with a .etl extension should be created in a path like this:
View the trace:
1. Launch Event Viewer (Start --> Run --> eventvwr.msc)
2. Action menu --> "Open Saved Log..."
3. In the "Open Saved Log" dialog, navigate to the .etl trace file you created earlier and choose Open.
4. When prompted to convert to the new event log format choose Yes.
5. Change the display name if you would like then click OK
6. You should see the trace information in event viewer.