从语法上看你的理解是有道理的，但实际上这里不用加上 being，因为 give sb / sth up for lost 是英语中的固定搭配，意思是 stop looking for sb / sth because you no longer expect to find them / it(认为……已没有被找到的希望；认定某人已死或某物已失去)。如：

The fishermen had been given up for lost in the storm but they have now arrived safely back. 人们本来认定这些渔民已在风暴中丧生，可他们却平安归来了。