Using a cure for an epidemic as blackmail, and just in time for my area to go into self-quarantine for COVID-19, wow I didn't think the Empress could get any worse.
If I were the Archduke, I would secretly spread rumors about the Empress or the Royal Family having the cure and are waiting to hand it out. I'd make sure it'd get heard to the area where the epidemic is affecting and places with lots of commonfolk to make it harder for the Imperial Family to quell the rumors. I'd word my rumor carefully saying that the Imperial family is currently preparing the supplies to make more of the cure so that I can't be held liable for saying anything bad against the Royal Family but just speculating. But of course, I'd have to make sure the wording gets changed a bit down the line to make it seem the Royal Family or Empress is guilty of something, so I'd just make the wording a bit vague so it can easily get twisted down the line. The more variations of the rumor, the better the chances that someone who try to investigate and actually find that they have the cure. Then they'll be forced to hand it out to everybody including the Archduke's wife.
Or not. I honestly have no idea if this would actually work and there's a high chance that I'd get rooted out before the rumor really spreads.