77 BU vs BB - to barrel river or not to barrel river

Villian is 21/17/7.5 over 1.2k and we perceive each other as solid thinking players.

Hero is 27/19/5.7, image is pretty clean other than I misread my hand a few orbits ago and got snapped off by a fishes raggy ace on river(thought i flopped broadway)

Cake Poker No-Limit Hold'em, $0.20 BB (6 handed) - Cake-Poker Converter Tool from FlopTurnRiver.com

UTG ($47.37)
MP ($9.86)
CO ($20)
Button ($54.79)
SB ($32.89)
Hero (BB) ($19.14)

Preflop: Hero is BB with 7, 7
3 folds, Button bets $0.54, 1 fold, Hero calls $0.34

Flop: ($1.18) 6, 5, 4 (2 players)
Hero checks, Button bets $0.99, Hero raises to $3.15, Button calls $2.16

Turn: ($7.48) A (2 players)
Hero bets $4.35, Button calls $4.35

River: ($16.18) K (2 players)
[color=#CC3333]Hero ???

Total pot: $38.38

I feel like c/r is a better option than c/c -> donk turn or c/decide because it allows me to represent more hands on turn and river, i feel like the turn is a perfect example of this.

turn sizing seems bad now that i look back, because when he calls flop im fairly certain he has a set or some combo draw, and pricing him to draw to board pair is not what i would do with a value hand.

so question-> does villian fold a set on riv if we jam, i know hes never folding a flush