Is it OK to join a Yom Iyun for Rebbis in Yeshiva, if the class is being given by a woman?
The answer below is in general terms only and not about your specific situation.
Under ordinary circumstances, a Mosad would/should always ask local Rabbonim about such matters and share with all the teachers the identity of the Rav so everything should be transparent.
Where a specific teacher is uncomfortable with a policy in the Mosad, he could be given the option to discuss it with the Rabbonim of the place/Mosad, thus avoiding anarchy.
Bichlal, it’s Kedai to allow an individual to follow his personal religious preferences and not try to force upon him behaviors that are inconsistent with his conscience. This is particularly relevant when we’re dealing with somebody whose formal job is to instruct in religious matters and inculcate religious values to students. It doesn’t make sense to try to tell him that he must operate against his own religious conscience.
Furthermore, it would appear that a religious individual who operates against his very own conscience and religious convictions would not be suitable to be employed as a religious functionary. It is noteworthy that even secular society recognizes the concept of “freedom of conscience” and/ or “conscientious objection”. (Cf. Heichal Menachem I p. 61)
As for the specific matter at hand, certainly, is not proper for a religious crowd. Indeed, many Poskim advise against this. Some even forbid it outright.
However, if the audience is not religious, and the purpose is to teach them Torah, there could be some room for leniency in very specific scenarios, and one should seek guidance from a Rav.
Nonetheless, it’s hard to say that it is absolutely forbidden according to Shulchan Aruch, though as mentioned some Poskim do forbid it. Certainly this isn’t Midas Chassidus and inappropriate for a Chassidishe Mosad.
I urge you to discuss with the Mosad and have them reconsider their decision, instead of creating a situation where everybody will do whatever they want and undermine the formal position of the Mosad.
See Shaarei Yosher III: 21. Shevet Halevi 3:14. Mishnah Halachos 5:218.11: 58. L’horos Nasan 1 EH 60:5. Igros Moshe OC 5: 12. See also ibid. 2:36.