There are a number of great reasons to go to a professional mediator. Among these is that mediation is inexpensive relative to litigation, it is as private as the parties involved let it be, and it can therefore be less "pressurized" than court or public donnybrooks.
The parties have to agree on the mediator, where the meetings will be held, and to be bound by the results. Payment to the mediator varies, but is reasonable, and usually has some relationship to the participants' abilities to pay.
Mediators can be anybody the parties can agree to, and there is the Federal Mediation and Conciliation Office available to help set it up.
However, the parties in conflict have to be willing to settle. Without the desire to end the conflict, the mediation will be interminable and produce nothing. There must be an incentive for each party to take part. That is, neither party should believe they are negotiating for their life.
If these organizations believe they are fighting to the death, to determine which will be the ultimate PW organization, neither will wish to enter mediation. They will each believe that fighting in the public opinion arena will be preferable to perishing after mediation.
Similarly, the party that most expects to win in litigation will believe that ultimate vindication in court may be the only way for either to survive. Again, a reason not to be mediated.
I hope that some sense can be brought to this dispute, but it seems that the history isn't promising.