Using the annotation functions of an external viewer means your annotations are not accessible to Zotero search
and I think that along with the desire to annotate anything that appears in the browser is the real issue.
One is working with any pdf reader which copies the highlighted text to the comment field of the highlight (as far as I know only Adobe Acrobat)
It also may be relatively trivial to read PDF annotations and display them as (probably exclusively read-only) Zotero notes. They're normally stored in the clear in PDF files, no?
I know, it's pretty ugly but hey I am usually just doing statistical programming...