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.