Christopher Pike knows a lot less than we, the viewers, know. We know he isn't being told anything about the Mirror Universe. All he knows is that Capt. Georgiou was believed dead in the Klingon war, and is now alive with Section 31 and acting unlike her old self. However, Pike DOES know about the Voq/Tyler situation. This was confirmed during his talks with Burnham who insists Voq's crimes don't represent Tyler. Pike knows Tyler is an officer who was believed killed in the Klingon war, was found alive afterwards and passed multiple medical scans, but then revealed to actually be a Klingon disguised as Tyler. Although Tyler's memories transferred, Voq's personality could have been reactivated by code words and that was only shut down because L'Rell wiped the Voq identity from Voq/Tyler. Now Pike sees another officer, Georgiou, suspiciously alive after being thought dead. She's acting strangely. Section 31 has already recruited another Klingon disguised as a human, Voq/Tyler. Why wouldn't Pike think the same thing happened with Georgiou? The Klingon spy program had 2 infiltrators, and Section 31 took them in. Mirror Georgiou's antics are just the Klingon personality breaking through once in a while, which Section 31 wasn't really able to erase. I'm surprised Pike hasn't outright said he thinks Mirror Georgiou is another Klingon surgically transformed into Georgiou. He already knows there's a precedent. The only reason he wouldn't think so is because the writers are too focused on how the audience (the viewers knowing about the Mirror universe) think and not how the Pike character himself would think.
And I think that's fine in this circumstance. Pike said enough for us to know that he recognized there was a difference, and there's really no reason to have Pike talk more about it unless it becomes a plot point in the future.
The story that she was presumed dead, but actually wasn't was spread Federation wide - as is the current post was story that she has retired (which Pike also mentioned). Neither story is implausible; and Pike has a habit of believing Starfleet Command until he has verifiable evidence to the contrary. Remember he DID bring this up with Burnham who replied to him: "It's a longer conversation..." and basically saying she would tell him later; so yes, while he has a suspicion something is up, he's probably not going to think about it further until Burnham drops the details; or another situation demands it. He (and the Admiral) was adamant he DIDN'T want Georgiou as the liaison officer from Section 31 aboard Discovery; so yeah, he has suspicions, and he's not dumb.
Pike said that he was "briefed on the classified details" concerning Lorca, so if he's aware of the mirror Universe, then more likely he would suspect that rather than her being a Klingon. But, as others have mentioned, he has no reason to disbelieve Starfleets official story at this point.
I think Voq/Tyler only passed the medical scans because it hadn't occurred to anyone to look for a Klingon whose internal organs and structure had been altered so substantially to match the superficial changes to the outside. They were used to spies simply getting their faces and skin changed to look human but still being internally Klingon, so a casual security scan would still detect them. Now that Voq/Tyler was known about, they would know to go through with a full body scan down to the cellular level of anyone seeking a job around high security ships and projects. Maybe that's why McCoy was able to identify Arn Darvin so easily (once the Tribble tipped them off). He passed the more casual scans of K-7, since it wasn't a high security station, but McCoy ran a deeper scan on him. That, or Darvin was only the superficial kind of disguise that just changes the outside.
I think the DSC doctors detected absolutely everything there was to be detected. It's just that what they saw was consistent with two things: a human mutilated to near death, and a human mutilated to death and then wrapped around a Klingon. That there would be fundamental differences between a cubic centimeter of human and a cubic centimeter of Klingon is a real-world conceit. In Trek, all aliens have DNA and are the result of the same project by superaliens who wanted to fill the galaxy with reproductively compatible sapient humanoids; the differences are in coarser things, such as the number of lungs or the shape of spines or whatnot. Timo Saloniemi
That makes me wonder, I always assumed Lorca kept a pet tribble to tip him off to disguised Klingons - I guess that he never had Tyler in his ready room, or Voq’s transformation was substantial enough to fool the tribble.
I think he would figure section 31 would have the same suspicions and trusts they would have vetted her.