I don’t believe that there is currently any way to export to BibTeX and have either the tags or folders be preserved as keywords in the BibTeX output. This would be useful when writing annotated bibliographies in which topics or subtopics need to be grouped.
Just select a label or folder and choose from the drop down menu “Export > Export to BibTeX”, it should do exactly what you requested. I have attached a screen shot for illustration.
This does work. I wish that copying individual or small groups of references from the “cite” button would do the same.
I would also like to export tags connected with each paper and their pdf either in bibtex format or, just plain text format with filename on my cloud along with all tags associated with each file.