I've been here with my soccer team in the last season. Feeling let down by top brass because we can all see a problem that they seem utterly unwilling to address, worse in this case as they actively appeared to stop their employees from carrying out the job they are paid to do.
Getting their attention requires mass participation, not online but in the stadium. I could be wrong but I suspect we are a fraction of a percent of Nats fans who post on here and I doubt all of us are united on this issue as well! I also can imagine, though happy to be wrong, that a lot of fans have virtually no interest in the rumours and machinations going on behind the scenes, they just turn up to watch baseball.
With these factors in mind I should be amazed that any sort of fan power could be brought to bear. But if it could, then hitting them in the pocket is the best way. Mass boycott of games or don't use concessions. Stop buying merch and actively avoid using products/services provided by sponsors.
Even then, depending how bloody minded the owners are (I suspect, very) it could make no difference.
As you suggest, The entire coaching staff and Rizzo and the scouting department quitting enmasse would probably provide the quickest answer, but they won't be doing that and we have no influence over it.
But don't worry! We signed Edwin Jackson today!