How do you even pick a therapist in this situation. I mean as polarizing as this topic is today I'm sure you can easily find one to support both of your views. How will your wife handle a therapist who is pro vaccine. Probably about as good as you'd handle one anti vax.
The therapist's personal views on vaccines won't matter if the therapist is even halfway competent. This isn't an issue, a properly trained, ethical therapist is equipped to deal with literally any conflict from a non judgemental position. Being non judgemental is the cornerstone of therapy. That said, being non judgemental is also the hardest part of the job, and a lot of therapists fail. I think it's especially hard for the faith-based therapists, because so much judgement is built in to a lot of sects of various religions.
It still stands though, that a properly trained therapist should be able to provide marital counselling without taking sides, period.
Now to OP:
As other have pointed out, your issue isn't the vaccine, the vaccine is simply the conflict that is revealing the underlying fractures in your marriage. Your marriage was always vulnerable to this form of conflict, and covid has revealed how deep the conflict goes.
Try to conceptualize the vaccine as the thing that's revealing the problem, and not the problem in and of itself. From there, put your energy towards trying to understand what the problem actually is.
You talk about feeling embarrassed by her. Why? What exactly about her behaviour is making you embarrassed. Ignore the subject matter, and focus on what it is about her that is driving her to behave in a way that you can't respect. What about her made her capable of behaving in a way that embarrasses you?
You feel emotionally manipulated. Has she always reacted this way in conflict? Have you ever felt like you can trust her with your vulnerability in a conflict? What would you need in order to feel safe when in conflict with her? What do you think drives her behaviour? What does she need to feel safe in a conflict?
How invested is she in resolving this? From the sounds of it, there are some significant emotional maladaptations going on here. Is she willing to seek individual counselling as well as marriage counselling? Because marriage counselling can only address a conflict between you, it is useless for addressing internal issues that she or you might have. So if there are underlying individual problems driving conflict issues, then those need to be addressed in parallel with the marriage counselling.
In fact, if I get *any* sense that the individuals need work, I recommend solo therapy *before* couples counselling.
Again, I repeat, try not to let the content of the conflict distract you. You're focused on it because you have a timeline to get the vaccine. Well, at the end of the day, that's not the issue. Get the vaccine if you need to, her feelings will be hurt, and you will add that to the shit that you need to deal with in therapy.
It's only in your head that you think this isn't solvable. But what you are actually seeing is that this isn't solvable without more conflict. You can't avoid this getting worse. If you just accept that simple fact, then what to do about the vaccine is quite simple. Remember, the vaccine isn't the issue, the structural marital problem beneath it is. The vaccine is just revealing how warped the underlying structure is.
So don't worry too much about the vaccine. It's your body, it's your career, there's no question that you should get it. The fact that doing so is going to cause a massive blow up isn't necessarily a bad thing. It's unpleasant, but it isn't necessarily bad.
You can't find a peaceful way to resolve this because there is no peaceful way to resolve it. The dysfunction won't allow it. And the dysfunction isn't going to be magically healed in time for you to get your vaccine. So worsening conflict is the only natural outcome. The sooner you accept that, the sooner you can move forward with figuring out how you want to handle it.