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Your question is quite similar to what I am working on right now. I also need to store some data in rviz into a file. So here is what I'll do:

Those positions and orientations in the display panel in your screenshot are just child classes of rviz::Property. So a bit of a roundabout way is to create a simple panel similar to this tutorial with a single 'save' button. And use rviz::YamlConfigWriter class to save it as a YAML file.

You'll have to dive into the rviz documentations a bit though.