None of the online dictionaries I have consulted - Langenscheidt, leo.org, Collins, verben.de, dict.cc - give me confidence in their recommended translations of "arbitrator". An arbitrator is different from a mediator. Arbitrators act as private judges and issue OUR decision, usually legally binding, about the parties dispute; mediators work to get the parties to reach their own agreement. Nor is an arbitrator the same as an umpire or referee (sports or otherwise) or an arbiter - we hold quasi-judicial hearings, usually take evidence/testimony, and decide. Some of these dictionary sites and others appear to assume that mediators, umpires, and arbitrators are identical functions.
To avoid the reaction "Oh, Fußball!" I have resorted to "Nein, eine private Richterin." The dictionaries offer "Schiedsrichterin", "Schiedsfrau", "Schlichterin","Unparteiische Schiedsrichterin", "Vermittlerin". Which is closest to the American labor or commercial ARBITRATOR? Personally I like Schiedsrichterin with its reference to judging.