> I can now see the basic problem is that I want to capture from ` `` ` (two back ticks) to the end of the line, but only when there are not two consecutive quote marks (i.e., closing quote is `''`).

    "``.*?\\(?:''\\|$\\)"

or equivalently:

    (rx (seq "``" (minimal-match (zero-or-more not-newline)) (or "''" eol)))