Why did Vance underestimate the Major?
Did Vance actually tell Saanvi to eat an apple at her therapy session? Yes, he did. And she did exactly that. It still didn't stop her from fidgeting. Nor did she hide her desire to hurriedly finish therapy and to get as far away from the Major as quickly as possible. Everything about Saanvi's body gestures and behavior loudly screamed out to the Major, "The only reason I'm here is to tell you this lie, hope you'll believe it, and then get the hell out of here as quickly as possible." It was great acting on Parveen Kaur's part.
Dislike for someone doesn't mean you should underestimate them
Vance should have known the Major would see through Saanvi's amateurish attempt to con her. Not only has Major General Kathryn Fitz been chasing "enigma's" for 22-years, she has been able to continue to chase whomever she chooses to chase because she's not stupid. Vance said this to Saanvi about the Major, "She's a seasoned intelligence officer."
Kathryn Fitz is much more than a seasoned intelligence officer. She's Harvard-educated with a PhD in Psychology. Couple her incredibly high intelligence with her cold-blooded ruthlessness, and she instantly becomes extremely more dangerous than anyone Vance or Saanvi has ever encountered. Why would Vance even attempt to use Saanvi, an inexperienced novice, as the individual to "trick" and "trap" the Major? Did he really think his plan would actually work?
Planning to do something and actually doing it are not the same
Either Vance is getting old and forgetful or he's in over his head. He trusted Saanvi when she said she used fake research specimens. She never told him about the real specimens hidden is her home. Then too, he never asked about any of the research she had already conducted. The Major never asked Saanvi about the secret research lab in her home. But doesn't Saanvi live with her parents? Regardless, the Major now has everything. Including Saanvi's fake research specimens and her real research specimens.
Who's to blame for losing the research materials and information? Vance? Saanvi? Both? What did Vance gain by underestimating the Major? What benefit did Saanvi receive from her therapy sessions? At least the Major showed that she was a good sport about it all when she sent a sarcastic "thank you" message by leaving Saanvi the "apple" in her vacated therapist office.
Btw, Vance should now know the Major had Saanvi under around-the-clock surveillance. How else would she have known about Saanvi's secret lab? One of her agents had been ordered to observe what materials she walked in and out of her apartment with, e.g., lab rats and cages? She also probably had Saanvi followed. There is a strong possibly Saanvi led the Major straight to Vance's secret hiding place.
If Saanvi was an important enough target for the Major to come out of the shadows and personally expose herself, then Saanvi was definitely important enough for the Major to keep close tabs on. Saanvi's apartment was also more than likely bugged. If Vance was the Major wouldn't he have made sure that Saanvi was always within arms reach?
After all, Vance himself is a seasoned career intelligence professional. When only using a flashlight in complete darkness, he's the one who discovered the tiny fresh bloodied bandage at the abandoned Red Door facility ... a facility which appeared to have been abandoned for years. Vance made a serious mistake by underestimating the Major. He can't afford to underestimate her again. If he does, it could actually cost him his life; a situation in which he won't get a chance to continue to fake his death.
Let's assume Vance underestimated the Major because he didn't know who she was; or anything about her background and experience. He probably thought that she was an amateur who was reckless and careless. But now he knows that he isn't dealing with with an inexperienced rookie.
Vance himself was military. So, now he knows exactly what skills and capabilities Major General Kathryn Fitz possess, and the resources available to her. Remember what Vance said before the Red Hook rescue mission, "Look. I was military, but I am gonna be standing behind a tactical unit. There’s no way I’m bringing in civilians." Vance made a serious mistake by using a civilian, Saanvi, in a reckless and failed attempt to lure the Major into a trap during a therapy session.
Did Vance underestimate the Major because he knew she was a woman? Or did he underestimate her because he thought he was more intelligent and experienced than she was in covert operations and national security matters. As stated elsewhere, there will be no second chances. Vance cannot afford to underestimate the Major again. She will be waiting and ready. Will he up to the task?