Honestly, if you refuse to do something on ethical grounds and then are moved around, you know what's going on. Most people aren't clueless to office politics and at that point it's your decision to blow the whistle or shut up and watch it happen in spite of your oath.

It's not like hospitals haven't contended with this exact thing for a very long time. The wills of surgeons/doctors and their hospital administrators do not always match up.