First, find out if the court itself has its own internal mediation program - i.e. a panel of volunteer neutrals to which it assigns cases filed in higher courts. Then you'll know if that resource is available, and you should be able to do that simply by calling the court and asking.
If the court has such a program, ask how it might be used for a SC case. SC cases are normally entirely handled on the day and date and very few states make provision for advance law and motion. If the court doesn't have an internal program, it's unlikley that it would entertain such a request under any circumstances, since it essentially amounts to ordering the case out of the court's jurisdiction - like ordering the parties to abide by a contractual arbitration clause.
A more common formula for SC courts is to use volunteer mediators in conjunction with the proceedings. You can also learn whether your court does this by calling in advance.
One thing to remember is that mediation is a process to which both sides must agree voluntarily to participate. Whether your court has a program or not, there is no guarantee that the other side would meaningfully cooperate.
There are profound advantages to mediation, not the least of which is that the parties can agree to a resolution that the court could never order. This is especially true of SC courts, where judgments tend to be confined to money judgments, when you want/need something else or something that is beyond the power of the court to order.
If you are committed to the idea and the court does not have the resources, one thought would be to contact the other side and suggest using an outside mediator (there are commonly community resources for a nominal fee, if not for free) in advance of the hearing. If the mediation does not resolve the dispute, you proceed with the hearing. If you feel that you could not successfully make this proposal yourself, find a neutral willing to serve and let him/her make the contact and convene the mediation. Trained neutrals are accustomed to this being a part of the process.