You seem to have a double stress: You are worried about your stepdaughter and you and your wife aren’t on the same page. I’m sure this is very difficult for all of you.
As much as I’d like to help, your intention of looking into the local hospital program is a better bet. Therapists there can help you all figure out what your stepdaughter needs. I suspect they will tell you that you’re right that she eventually has to learn the social skills she needs to fit into the social world at least part of the time. I also suspect they will tell your wife that she is right to choose her battles carefully. Hair grows. Clothes can be changed. For now, it might be more important to focus on helping the girl learn how to manage her feelings and how to use that genius IQ to help herself.
I urge you to take her for a thorough mental health evaluation and then listen carefully to the proposed treatment plan. By attending family sessions, you will demonstrate to your stepdaughter that it’s okay to reach for help when you need it. You may also learn more about what is distressing her and what you can do to be more supportive. By working together, you and your wife can provide external structure and safety until she learns how to provide it for herself.
Your stepdaughter is lucky indeed to have such a concerned and caring stepdad. Love is just as important as formal treatment in the outcome of situations like these. Do keep loving her. Do keep working with your wife to be more together in how you approach the problem.
I wish you well.