become用于进行时态是很正常的用法。如：
It was becoming very dark. 天越来越黑了。（《英语用法指南》）
It’s becoming a serious problem. 它正在成为一个严重问题。（《张道真英语语法》）
Billlards is becoming more and more popular. 台球越来越普及。（《朗文英语语法》）
Jamie was becoming annoyed with me. 杰米变得对我不高兴起来。（《张道真英语用法》）
你的句子完全没有问题：
It is rapidly becoming clear to me that I’m not suited to being a teacher. 我很快就明白过来，我不适合当老师。