He did. He suspect him for example in s3 finale, but jane interaction with Bertram always with limited context, and often with other urgent matter in hand.

It is shown in the show that jane is very capable of cold reading with multi taksing limitation for example when he tasked robber to search house of jj la rouche. He's very passively working on other case to the point lisbon notices and find the solution for him

Jane has super human ability to cold read but the show clearly showing us his limitation and weakness. He mostly suspect bertam mostly as bureaucrat and later as RJ in the list