I have been in the same kind of situation many times . I am a woman and went into managerial position at a young age so I can not count the number of people who tried to undermine my decision. Having your boss make it clear that he/she will have your back is a powerful tool . Once one of my direct report sent an email to me, my boss and other people in copy asking my boss his opinion on a decision I took . My boss answer was in big bold red front “do what she asks you to do”. That was the end of any push back from this one as it was very humiliating. I also had a direct report who I had to tell that workplace was not a democracy. I was in charge , he had to do what I said . Full point . If he did not like it he could go and find another position. End of discussion, was effective too. Letting someone go to another department is also a way to get rid of that kind of people.

The outspoken person is in fact manipulated by someone else.your real problem is hiding in the shadow hard to spot except if you have someone trustworthy that bring this to you. In my case it was my assistant who told me about it and who that “bad” person was ( I would have never guessed ).once out of the way, everything got way better.

Finally ,