Moshe,
Do as your wife wants for now. Don't go Tuesday night. Show her she can begin to trust you again. Set a boundary with her that you will not go to meetings when she needs you to help her. You need to find out exactly what and when that is and schedule around it so you're both satisfied. Also, how does your wife know what you discuss in the meetings? Is she assuming or are you telling her? Don't tell her any details about what is discussed. You're not violating HER privacy, you're healing from your shame, guilt and anger. You need to run your program and let her run hers. She needs to go to S-Anon meetings and see how others deal with their husbands. But if she wants to know the truth, have her call my wife. If G-d forbid you had a fatal disease would your wife say you shouldn't take medicine because everyone will know your business? I would hope not. Meetings are the medicine to recover from this. PERIOD. You can manage for a week or two. Make some phone calls to people in the meeting every day, or call me. You can still journal, read the book, work the steps, talk to your sponsor. It is more important to spend time WORKING the steps than going to meetings.
It is not worth breaking up your marriage if you can switch meetings. Your recovery is priority #1 but not at the expense of divorce. I know your sponsor disagrees, but he should be willing for at least a few weeks to talk with you each day and maybe meet during a time that your wife does not need you.
Let me know what happens.
Elya