Pam, you say "clearly she respects only the authority which can fire her", but correlation does not equal causation.
From your remarks it looks as if she is capable of doing the work you're asking her to do, and responds well to other people's requests to do it.
You say that she's getting more argumentative and disrespectful. How are the rest of the team responding? My experience of psychology tells me it's less likely to be the threat of firing and more likely to be personal.
It could be that she likes the team more than her managers, and feels free to offer up her opinion. Or, perhaps she feels that her opinion isn't listened to, and is seeking an audience for it, making her appear defensive and argumentative.
In either case, one solution might be to simply listen to her arguments and show that you've heard and understood them by playing them back to her. If you can do this, she will feel respected, and will probably accept any decision you make, even if it goes against what she would have done.
Right now, you have classified her behaviour as unprofessional and disrespectful. That you say it might be cultural too tells me you could be suffering from a fundamental attribution bias - that you consider her behavior to be innate in some way, rather than caused by something in her environment. Please take a moment to consider what she might be responding to, and how you can provide a different environment in which she will behave more constructively.