It's hard to say if Burke knows it's the MC specifically or not. It depends on how much Burke has.
The clip that the DIKs saw has Jade calling out the MC's name, and Burke is looking at that same screenshot. If he has the full video clip, he knows it's the MC. But if he's missing the audio portion, or only has screenshots, then he might not know who it is.
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[EDIT: Fun fact: the MC's name in that clip will not change even if you update the MC's name via the phone.]
Personally, though, I assume he knows. Unless someone specifically wanted to protect the MC, there's very little reason to give Burke this information without giving him all of it. It makes certain Burke recognizes Jade, and catching her with a student makes it even more damaging.