If I understand correctly, the running heads are the book title on the recto and the author’s name on the verso?
If so, I wouldn’t even bother with text variables. I’d just type it in the running head box on the master page.
If the author’s name doesn’t appear until the very last page, then InDesign is doing what it’s supposed to do: Not putting the author’s name in the running head until it sees it. The only way to force it to see it earlier is to also have it on the first page as well. Since you don’t want it to print, you can always highlight the author’s name to print as none or white.
I personally recommend using character styles for running head variables (they can be nested or in unique situations, the title can be highlighted physicall). The character style is empty of course.
As of now, InDesign can’t go to the end of the book, find the variable for the author’s name, and then backtrack to make it a running head. It won’t make a running head until it actually sees the variable.