No-Limit Hold'em, $0.25 BB (6 handed)

BB ($64.44)
UTG ($25.70)
MP ($15.70)
Hero (CO) ($25)
Button ($9.65)
SB ($16.68)

Preflop: Hero is CO with 6s 6c
2 folds, Hero bets $0.75, 2 folds, BB calls $0.50

Flop: ($1.60) 4h Ks 6h (2 players)
BB checks, Hero bets $1.20, BB calls $1.20

Turn: ($4) Qh (2 players)
BB checks, Hero checks

River: ($4) 4d (2 players)
BB bets $2.25, Hero raises $6, BB raises $36.25

Villain in this hand is a reg nit. When he calls my flop bet I know for sure he has a piece. Could be a set of 4s or a FD or Kx. Turn brings the flush, so I check behind. River boats me up. He then leads. To me this is either a flush trying to eek a little bit of value out of any Kx I might have, or better. I raise thinking he'll call with the flush. However, he shoved. Looking at the board and knowing his ranges (he has K4s like never), I am BEAT right?