What about connecting multiple facebook sources? Profile, page and group? Is that in plan too or do you consider it an overkill?
Jetpack doesn't allow you to publicize to multiple pages from a single account, but you can add a new connection, to a new Facebook account, to work around the problem.
Start by going to Facebook, and log out of your account. Log back in with another Facebook account, listed as an admin on your Facebook Page. You should then be able to go back to Settings > Sharing in your dashboard, and click on "Add a new Connection" to create a new Publicize connection.