People who could be the headmaster:
Lara: Headmaster is future Lara (The voice recorder bit makes me think that current Lara is not the headmaster unless she is doing it for someone else to find). Future Lara is manipulating current Lara the same way current Lara manipulates. . . everybody. Headmaster Lara doesn't let current Lara kill Gribbs because Gribbs is important to Headmaster's plan even though Headmaster knows how much Lara wants him dead. Current Lara is so freaking confident because she is being fed info from her future self.
Daramount: Either it's a future self or she's a really great actress (leaning towards the latter since she has no "gifts" and you need gifts to time travel, right?). She whips herself in the greenhouse to gain Lara's sympathy and help (it's why there's only one shadow in there). "Headmaster" is a title that is passed down: when she was a child, Gribbs/Albert/Handsome bearded teacher/random teacher/security guy was the headmaster, then passed the title and power on to Daramount (It can't be Dagney, or Daramount would treat her with more respect, Nine is too young, and she wouldn't let guards in if it was Abraham). Cons: She seems way too worried about the headmaster to be him, but then, acting.
Gribbs: Gribbs is mysteriously immortal. As one of the more background figures, he is able to be overlooked.
Albert: See Gribbs, but no immortality observed.
Abraham: Mysteriously immortal. Used to be associated with academy personnel. Playing both sides against each other. Con: Who pulled him from Coney island with Ike?
Oliver Simon: Crazy smart, breaks rules of nature. "Capture" is cover to visit Academy incognito. Ian was created to infiltrate Abraham's camp.
Miss Richmond: Like Simon, but infiltrates camp herself. Vanessa is a pawn.
Ike: (future version) Irina thinks that killing him will fulfill the truants' mission (after failing to kill the headmaster). He's related to Abraham (who's suspicious). He's callous. Premonition powers.
Zoe: Super mysterious character with a lot of loose ends. Has powers. Maybe headmaster is her alternate personality? Or Headmaster is like the Dali Lama and is reincarnated into new bodies?
Casey: Already has time traveled, a really good liar. Intentions unknown. Willing to manipulate others for own purposes.
David (Casey's son): Ominous name, unknown story, probably has powers.
Tom Reed: Involvement with car crash? Suspicious influence on Casey.
Hunter: (future version) Known to build weird time contraptions. Undefined powers. Brain thing.
Jade/Dr. Ellsworth: Already demonstrates cross-time communication/travel. Association with the cylinder.
Junisao: (future version) was deemed important to academy. Now super pissed and unpredictable. Would do anything to get brother back.
Ian: (future version) Ian isn't *completely* human, maybe once he grows up he can reach through time. Callous enough to be headmaster.
Irina: (future version) Powers, Daramount's sister, resents Abraham. Could explain why after trying to kill the headmaster, she's now going after Abraham and family.
Guillaume: Time powers, willing to manipulate Hisao. Went to Abraham out of a "calling".
Akiko: Innocent facade hiding devious mind? Could be headmaster running things with faith to higher plan. What was father's association with Academy?
Akiko's father: Suicide was implanted memory. He's the mastermind.
Vanessa: Would do anything to save Brandon. Powers. Older Vanessa looks tough.
Sentient plants: Explains the greenhouse. Vines used for whipping.
Computer: Time travlling A.I?
Salem Guy: Is cruel, knows about future events. Ties to academy.
Roy: Observed Zoe's initiation. Has ability to escape from cage. Nobody suspects the ram.(Man, Jun would have been in trouble if he had completed the ceremony. . . )
Basically, *anybody* could be the headmaster. Your Lara theory has some merit, but I think that current Lara as headmaster is problematic. Current Lara obviously hates Gribbs and would get rid of him. Lara is well liked by the students and could probably get them to do what she wants. I would propose that she's faking her dislike and using Gribbs as "bad cop" except for the private speech in the hospital (unless that scene was purely for Akiko's benefit).
"Headmaster" existed when Lara was a child (unless it was a time-travelling Lara who later made the hand-off to current Lara). Why is she taping a message to herself? She *runs* to save Daramount from the greenhouse, if she was the headmaster, she wouldn't need to.
The whipping: Daramount says "I tried to explain, but he's so angry" implying that she communicated with the Headmaster inside the greenhouse. It implies one of [Lara has a really good "headmaster" disguise and the ability to teleport (since she arrives running to the greenhouse), she has a permanent sock puppet headmaster (since it's someone Daramount recognizes as the headmaster), Lara has disguised a machine as the headmaster (is this a persona that she also inhabits herself? Or just a non-meat sock puppet?), Daramount recognizes the cylinder as the headmaster (but the cylinder seems to always be in the basement, and would Daramount call the cylinder "father"?)]. We know from issue 23 that whatever's in the greenhouse can be damaged. Even if Lara had arranged for Irina to kill her headmaster-self, it is unlikely that Lara would have the detailed knowledge required to have a "critically hurt headmaster" set up for Lara and Georgina to find.
What are you proposing about the Lara-Vanessa interaction? That 11 year-old Lara was already the headmaster and knew all about her mother? Then she's manipulating Daramount when talking in the bunk bed. Did she have fore-knowledge of Vanessa's arrival? Why such an elaborate ruse to capture Vanessa? 25 year old Lara already has Vanessa's trust. However she says to "wake my father" which implies that other people are interacting with the Headmaster (without knowing it's her), and that the headmaster is a separate person. Dagney also tells Lara that her father is proud, implying that Dagney has talked to the headmaster, and it's not Lara. Also, if Lara is (will be) the headmaster, why does she need to impress him?
Lara is definitely sneaky and untrustworthy, but there's nothing making her an outstanding candidate for headmaster. It appears that none of the Laras we've seen so far is the headmaster, and they are unaware if a future version of themselves is headmaster (even though the Laras have a spooky trans-temporal communication system).