I refer you to Article 2242 of the Catechism of the Catholic Church. Basically, the short paragraph states, like mentioned earlier, we never have to obey disobedience to God; when to obey man(or woman, in your case, it seems, maybe several), means to disobey God, then we do not obey man. Yet, we still give to Cesar, what is Cesar's, meaning, we still contribute to the common good, respect the moral order, the fundamental rights and dignity of persons along with the Gospel Teachings as always.
This is what we all discern every day, yes, even the Holy Father, when dealing with his Vatican staff, from his personal secretary, the Secretary of State who usually draws up his itinerary, to his chauffeur of the Popemobile. As a Parish Priest I must discern if I should listen to a Diocesan Policy or how it is applied by a particular person or Superior as well. All is not necessarily simple, since we are living in a Cloister either. On the contrary, Religious have more Constitutions with their Rule to consider before saying Yes and No. More is given to them, more is expected proportionately. The difference is, they made they have made a life-long committment, as of perpetual vows to live in that same Community. To have to look at that same face I had a disagreement with, day after day, is very difficult, as in Marriage, or in a family. As a Layperson, the person can leave a position without the same implications exactly. You have more freedom in the world, but the price is more responsability and discipline to seek guidance. No bell tolls to tell you what to do, in that sense. For me neither, though I would like to unplug the phone sometimes...
At work, if we are asked to do something sinful, or course the answer is No. If it is a matter of common sense and work ethic, that is not a grave threat to the common good, moral order, salvation of souls, or Gospel values, then it is matter of possible fraternal correction due to ther irresponsability, refusal to assume the consequences of their actions. The way to apply it varies according to protocol. First, face to face, then with witnesses, meaning consulting others by right, as in their Superior or colleague as to how to work with them. Then, you consider your options and their risks, transfer of department, demotion, change employment etc. whatever that may entail. This is the cost of witnessing to Christ. We've all been there, those who are true disciples.
I had to make such decisions before being a Priest, as a Catechist in a Catholic High School and in other employments. I thought it would stop in the Church...gets worse! and more challenging(and fun after a while) and there is more at stake. Transfer of Diocese?of Parish?Be sent into exile?If need be, but never, never leave the Church because she is made of sinners, that is for sure. What you are describing is martyrdom-which means witness of course. Feel free to send me more specifics, if I can be of help.