Short answer: No.
The only way this could currently be done with WebExtensions would be to ask the user to download and install a native executable, which the add-on could communicate with using nativeMessaging. But that seems like a total overkill in your situation. And you would have to detect the privileges of a different process than the current one, one yo don’t even know the process id or anything else than maybe a window name of.
Mozilla plans to add abstract APIs for specific tasks, but I highly doubt this will ever be among that.
So you can officially kiss your add-on good by once the everything except WebExtensions stops working in a year or so. But on the bright side, all other add-ons that do more exotic stuff will stop working too : )