Honestly, if you have anxiety problems this career is not for you, period. Acting is a VERY VERY VERY stressful job, even at times when you don't actually have the job. You will not be able to handle it: The rejection, the judgement, the deadlines, the meetings and networking, the long waiting, the uncertainty, the other actors, your acting teachers, the orders from the director, the audience when on stage or on camera, perfectionism is you're also a perfectionist, the disappointment and self-doubt, the loneliness, the lack of money, and so on. And for you specifically, the wig and what might happen with or without it. There's too much to cause stress. Even if you do get help and get over that problem, it'll always be there as a part of you. You'll always be prone to anxiety. So your question is irrelevant.
Rethink it. Health first.