Which gets to the point. It tends to work in the other direction. I have heard Jewish chaplains use specificly Jewish formulae and never get called on it. Much gets past these days, but strictly speaking, all mixed prayer is problematical. Ecumenicalism tends to shade into indifferentism. So whoever wrote the job description, whoever has made this rule, has taken sides in a theological dispute. To the Christian the name Jesus is the only name we use in addressing God. Jews have felt insulted by the statement of some Christians that their prayers are in vain. One can argue, as I do, that such a claim attempts to place limits on the All-mighty, and Ignores the historical fact that many Christians continued to attend synagogue even after the destruction of the Temple.
IAC. Christians who pray in the company of non-Christians are already making a concession, a bit like the non-Catholic who attends mass, or a Catholic who attends a Protestant service. Now, what gets to decide the matter? The guy with the money? Is that right?
Of course. Would it make sense for a non-Catholic to take such a position at a Catholic organization? Of course not, they want Catholics in such roles.
Whether you can deal with this multi-religion preaching is a personal matter of your religion. You can't, and that's fine -- those are your sincerely held personal religious beliefs and I will not deride them. But if you are not the type of Christian who can deal with this stuff, then don't take a job where it is required.
And as was said, you are free to avoid the organization if you don't appreciate the policy.