I'm having a hard time believing that the federation has a test for this sorta thing, a test to find if you've been brainwashed, bit doesn't have a policy that requires you to notify your captain if you find someone may have been.
I mean we've already entered into a nerdpicking moment of trying to apply practical real world ideas to a sci fi fantasy show, but it just doesn't strike me as something the federation would overlook
Well, I have no reason to believe that Dr. Culber would not have notified Lorca if Tyler had not killed him so suddenly.
Keep in mind, he has to balance multiple ethical issues. He knows that the second that he reports suspicions that Tyler will be grounded, and so far he has seen no evidence that Tyler posed a danger to anyone. I think that it was perfectly reasonable for a doctor to want to do a few more tests or at least talk to the patient for a few minutes before doing something that he knows will have severe consequences for someone's career, especially since they are currently trapped in another universe.
The patient had come to him seeking help, after all. The initial tests that had been performed did not set off any alarms. From a practical standpoint, if word gets out to the crew that Dr. Culber will run to the captain anytime something looks odd to him then many people will hesitate going to the doctor out of fear that something will happen to them and possibly not get treated in time for a problem.