I Think checking turn is better here.

V was a fish with 50%vpip who i'd caught bluffing before

No Limit Hold'em $0.01/$0.02
888 Poker
5 players
Formatted by pokercopilot.com: Poker HUD for Mac and Windows

Stacks:
BTN - Hero ($3.01)
SB - SB ($0.94)
BB - BB ($1.00)
CO - CO ($1.02)
BB - BB ($0.80)

Preflop: ($0.05, 5 players) Hero is BTN with Tc Ah
1 fold, BB checks, Hero raises to $0.08, 2 folds, BB calls $0.06

Flop: 5s 5c 7h ($0.29, 2 players - Hero: $2.93, BB: $0.70)
BB checks, Hero bets $0.09, BB calls $0.09

Turn: Td ($0.56, 2 players - Hero: $2.84, BB: $0.61)
BB checks, Hero bets $0.18, BB calls $0.18

River: 2s ($1.10, 2 players - Hero: $2.66, BB: $0.43)
BB bets $0.45 (all-in), Hero calls $0.45

I don't think checking turn is a good idea when so many river cards are scary.

I'd size a bit bigger OTF, although V is ridiculously shortstacked and I'm never sure what to size in these situations.

I don't understand the pre-flop action as you've explained it here, though; how is BB checking before you act on the button?