for the sake of what reason tells him is fit to be done

for the sake of something为原因状语。

what reason tells him is fit to be done为名词性关系分句。其中，what=the thing which。what is fit to be done为嵌入分句，嵌入到reason tells him这个分句中，作tells的直接宾语。

for the sake of what reason tells him is fit to be done

= for the sake of of the thing which reason tells him is fit to be done

在这个结构中，定语从句which reason tells him is fit to be done修饰the thing. 在这个定语从句中主语为reason（理性/理智）tell him为谓语+间接宾语，which为直接宾语从句中的主语。（嵌入式定语从句）。

有人将reason tells him理解为插入语，虽然不够准确，但确实能让这种定语从句解释起来容易一些。

本句理解的关键是，不要误以为what修饰reason。reason在此是理性、理智的意思，抽象名词，如用零冠词。what是is fit to be done的主语。which is fit to be done是reason tell him后的直接宾语。