不能加 is,因为前面是一个完整的单句,后面是逗号,说明这是一句话。加了 is 后,变成一个主语,两个谓语动词用逗号连接:语法错误。happy that ...整个是形容词短语,作状语,原因状语。可以转换成一个状语从句。
我举一个相似的句子: He couldn't sleep that night, excited that he had won the prize. = He couldn't sleep that night, because he was excited that he had won the prize.