Step 4 level 2 KK

This hand occurred early on so I had no read on villain. However, this is a step 4 500+35, so my presumption is that I wouldn't see overly loose play.

Seat 8 is the button
Total number of players : 10
Seat 1: HERO ( $975 )
Seat 2: gdlookn1 ( $1335 )
Seat 5: stang10 ( $990 )
Seat 6: eRpvCgOtAOej ( $920 )
Seat 8: vadka ( $820 )
Seat 9: RodTidwell ( $970 )
Seat 10: bingoken ( $1060 )
Seat 7: luvmyaceking ( $955 )
Seat 4: Devanaire ( $970 )
Seat 3: maxjan21 ( $1005 )
Trny:16292729 Level:2
Blinds(15/30)
** Dealing down cards **
Dealt to HERO [ Ks Kc ]
HERO raises [100].
gdlookn1 folds.
maxjan21 folds.
Devanaire raises [250].
stang10 folds.
eRpvCgOtAOej folds.
luvmyaceking folds.
vadka folds.
RodTidwell folds.
bingoken folds.
Your time bank will become active in less than 20 seconds. If you do not want it to be used, please act now.
HERO raises [500].
Devanaire is all-In [720]
Your time bank will become active in less than 20 seconds. If you do not want it to be used, please act now.
Devanaire: if u good u are a good player
Devanaire: fold
HERO will be using his time bank for this hand.
HERO calls [370].
** Dealing Flop ** [ Ts, 9s, Jh ]
** Dealing Turn ** [ 2h ]
** Dealing River ** [ 5d ]
HERO shows [ Ks, Kc ] a pair of kings.
Devanaire shows [ Ac, Ah ] a pair of aces.
Devanaire wins 1985 chips from the main pot with a pair of aces.

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I came out firing 100 and when I got reraised to 250, I of course went ahead and bumped it to 600.
A) Should I have just pushed here?

I was then reraised AI. 370 to call. At this point, I figured villain for QQ - AA or possibly AKs. I was further confused by the villain's comment that "If u good u are a good player fold." Without any particular read other than this comment and the way the hand proceeded, does anyone lay this down here and save 370? As a result of this, I finished 10th. I somewhat regret not laying this down as I almost "expected" to see AA, but I could not let go once I factored in the possibilities of QQ, KK, AKs, and a rare other. Thanks.