correct river size?

villain is call station here. seeing 90% of flops and 50% of showdowns with virtually NO aggression. what is the most EV bet size on river?


$0.05/$0.1 No Limit Holdem
7 players
Converted at weaktight.com

Stacks:
UTG ($8.30)
UTG+1 ($9.85)
MP ($6.00)
Hero ($7.55)
BTN ($5.15)
SB ($14.10)
BB ($9.90)
[MP posted $0.1]

Pre-flop: ($0.25, 7 players) Hero is CO :8d: :8c:
2 folds, MP checks, Hero calls $0.1, 1 fold, SB calls $0.05, BB checks

Flop: :ac: :8h: :js: ($0.4, 4 players)
SB checks, BB checks, MP checks, Hero bets $0.4, SB folds, BB calls $0.4, MP folds

Turn: :9s: ($1.2, 2 players)
BB checks, Hero bets $1, BB calls $1

River: :jc: ($3.2, 2 players)
BB checks, [color=#cc0000]Hero bets ??

raise pre.
Villain is BB = any two cards. Has either an ace or a jack, your read 50% of showdowns, further read (timebank here!) how often is he folding to a river bet?
I bet pot or push, probably push. If he has 9J, bummer.