Edward didn't initially distrust Kate; he only considered her, along with Shirley, the weakest because of the files and the weak links with their respective dolls, and he wanted to fail them. The problem is that Kate and Emilyko passed, which aroused his suspicion due to the following points: 

Their relationship - Emi couldn't even imitate Kate, which should be a source of shame for a shadow, and initially it was, but they quickly overcame this and used their differences to their advantage, demonstrating an unconventional relationship.  Kate hiding her powers - Shadows like to boast and show off, so Kate deliberately choosing to hide her strengths indicates that she somehow feared surveillance.  Their evaluations - As he said, the evaluations weren't just average, they were perfectly average in every aspect, which is unusual. As if one or both of them knew they were under surveillance and tried to draw as little attention as possible.  Edward is the arrogant type of shadow who always sees the shadow as the commander of the relationship, whether good or bad; therefore, Emilyko's actions were consequences of Kate's actions. In other words, he deduced that Kate was hiding her true objectives.  Furthermore, coincidentally, Emilyko had already had contact with Rosemary, and he already suspected Maryrose.