Is there a quick way to copy image metadata stored in LRC?

"Publish" and "Export" are effectively the same in this respect: you are making something that is visible outside of LrC's Catalog. When you Export / Publish out of LrC, metadata is embedded in the file produced reflecting camera and lens info - to the extent that this was not somehow stripped from the photo that was imported - plus further metadata (such as keywording) that's been applied within LrC. Besides reflecting the visual appearance changes applied within LrC.

Export settings include some 'privacy' settings which can suppress selected info, such as Location or Person labelling, from being included. But you do not have complete detailed control in LrC over which metadata fields are passed through, and which are blocked. If you wanted to suppress some particular metadata that LrC does not present an option for, while including general metadata, you can include a call to ExifTool with some suitable command-line parameters defined, as a post-processing step within your Export setup. As an example, say you wanted the photo's viewer not to see your own private Notes about the photo, ExifTool could be made to step in and clear that particular field, for each exported photo.  

Once you have exported, this metadata is held in industry standard form within the image file. Hence if uploaded to most commercial / free web galleries or else as an Adobe web gallery, it gets understood and can e.g. be used for filtering and searching and such. Also by clicking on whatever "show info" button is available, the metadata is revealed to the viewer of that photo.

If OTOH you are looking to create a separate text listing alongside some exported images summarising these same metadata, various utilities exist for that as a subsequent operation. Once the photos are exported / published outside of LrC, such utilities do not need to be Adobe specific in any way.