He is what is known as a “bellyacher”——he is always complaining about something.
不可以用who替换what。

这里，what 是“融合性关系代词”，它可以表示“人”，也可以表示“物”。它的作用身兼两职，既是关系代词，又兼有先行词。what 等于：

the person who...

the thing that...

本句的主语是He，表语肯定是“人”，所以，what is known as a “bellyacher” = the person who is known as a “bellyacher”。

who 不具备这个作用，它不是“融合性关系代词”。

He is what is known as a “bellyacher”——he is always complaining about something.
=He is the person who is known as a “bellyacher”——he is always complaining about something.
他就是所谓的一个“牢骚包”—— 他总是在抱怨些什么。