This isn't a full solution, but perhaps inter-tab communication would work? Use whichever tab is focused as the master, then other tabs send messages to the primary as they need work done. It wouldn't work when none of your tabs are focused, but perhaps that's an acceptable constraint.
That might work, but it's much more complicated and given the issues I'm already facing, I wouldn't want to rely on that communication actually happening (maybe background tab won't be able to send a signal to the foreground tab, since JS is apparently not running in the background tab).
