Fourth, present mediation as a routine part of your regular litigation plan, the same way you approach a deposition. Fifth, you can mediate from a position of strength. Sixth, the mediator can make the other side listen to reality when they won't listen to you. Finally, it is never an issue if court ordered or contract mandated. Attorneys can negotiate their own settlements, often using mediator techniques, but they cannot mediate a case in which they are an advocate.
Mediation requires the trust and confidence of both sides.
Opposing counsel are adversarial, they don't trust each other. Mediators have access to confidential information from both sides which facilitates settlement, attorneys do not. Both sides have hidden agendas. The mediator has none. Mediators are neutral, attorneys are advocates.