"Gentle jawboning" is lobbying. It's not necessary that the EAA do the continual self-back clapping that the AOPA does whenever something happens they want to take credit for.
You can try contacting your senator or congressman provided you didn't vote him out yesterday (or at least claim you supported him anyway) and ask if he could exert some pressure on the Secreatary.
DOT is where the politics are played. Once it gets to the Office of Mismanagement and Beancounting that approval should be relatively forthcoming. In fact, it should be a slam dunk for DECREASING federal budget impact as it will take a lot of load off Joklahoma City.