the most they would do to show their good faith is to wave t

这里是个隐藏主语.
先恢复成完整的、符合正规语法的句子：The thing they would do the most to show their good faith is to wave their registration card to the waiter.
所以这句话就是省略了主语the thing,把状语the most提到最前以起到强调作用.谓语还是is,宾语是to wave their registration card to the waiter.
当然了,把the most说成是主语也是没有错的,相当于“the+形容词”可做名词的用法.
两种理解都是可以的.