Poker Live tourney hand advice. AAvsKK

Okay, tonight was in a live tourney. $100 buy in, NLHE. 20min rounds, 5k starting stack.

My stack = 4.5k
Villain Stack = 8k
Unsure on villains playstyle, seems TAG, havn't had enough hands with him to see, tournament is only just over an hour in.
BLINDS: 75/150

I was on the Button.I get KK.
Villain in Early position raises to 400.
Everyone folds to C/O, who calls 400.
I 3 bet to 1200.
Villain shoves all in.
I call.
C/O Folds.
Villain shows AA.
Board comes dry so he wins and busts me.

My reasoning:

Villain raises to 400, only 2.5bb, when the C/O called, I wanted to reduce the field. (KK is good, but not THAT Good against alot of players).
I 3-bet to 1200 here and I feel like that's a pretty reasonable/typical 3 bet. I know it usually thins my range here though but I figured a 3bet would be enough to get him to fold any suited connectors/ JJ or lower. Maybe even AK.

When Villain shoved, I instantly put him on 1 of 3 hands. AA, KK, or AK. Knowing I had the other 2 kings I figured AA or AK. However my 3 bet plus his original raise of 400 put me 1600 chips into a pot with only 4.5k stack, so I felt like I was pot committed... So I called...and busted....


My Analysis:
After going over this in my head over and over again, I feel like my 3bet was fine because I was in position and I had the hand to do it, however I feel like when he shoved that maybe I should've just given him the pot.
I would've been left with around 25BB. Enough to make a comeback, considering I was on the button so had almost an entire orbit of "free hands".


Was my play correct? Was my analysis correct? any advice? This somewhat tilted me pretty bad and put me in a shit mood because I was only just over an hour into the tournament. I think I need to study more and stop making stupid calls.

Source.