I can't cite you specific companies. But I do know that everything at every company is usually on a case by case basis. And I doubt that any company would give coverage for "mental anguish" alone. Do you know how difficult it is to get medically necessary regular psych appts reimbursed?!
Usually you would need to have some type of dermatological problem; like severe rashes and/or infections to get it covered. And even then.... good luck.
But, as always, I wouldn't really worry about ANY of this until you have lost all of your extra weight. One thing at a time.