NL30 AA river?

On Game, $0.15/$0.30 No Limit Hold'em Cash, 5 Players
Poker Tools Powered By Holdem Manager - The Ultimate Poker Software Suite.

BTN: $75.19 (250.6 bb)
SB: $35.18 (117.3 bb)
BB: $47.69 (159 bb)
Hero (MP): $44.52 (148.4 bb)
CO: $19.24 (64.1 bb)

Preflop: Hero is MP with A♣ A♠
Hero raises to $0.90, CO calls $0.90, 3 folds

Flop: ($2.25) 3♥ 7♦ Q♣ (2 players)
Hero bets $1.65, CO calls $1.65

Turn: ($5.55) K♣ (2 players)
Hero bets $3.50, CO calls $3.50

River: ($12.55) 6♣ (2 players)
Hero bets $6, CO raises to $12, Hero folds

Results: $24.55 pot ($0.98 rake)
Final Board: 3♥ 7♦ Q♣ K♣ 6♣
Hero mucked A♣ A♠ and lost (-$12.05 net)
CO mucked and won $23.57 ($11.52 net)

Unknown.

I think betting the turn is ok.

On the river? I think betting again vs. a player with 64bb is ok. Size?
If the guy was 100bb (more likely to be regular) deep do you bet the river or just ch/f?