Solution
NI TDM Server Diagnostics Util is a tool that ships with TDM components for SystemLink. You can find it in C:\Program Files\National Instruments\Shared\Skyline\Diagnostics\NationalInstruments.TDM.DiagnosticsUtil.exe.
After launching this tool, you can manually configure the Level column to 6 (which is the maximum value) for any of the TDM components you want to capture additional information for troubleshooting. The Keywords field is used to control verbosity, you can set this field to 0xFFFFFFFFFFFF to maximize the verbosity of the output for the corresponding service.

The options in the top of the user interface can be used to select, load and save your templates which allow you to save configurations (like setting every value to 6) to avoid repeating the configuration later on.
The options in the right-side of the user interface can be used to start and stop the logging as well as saving the logs or clearing the output.