Poker Stars, $0.10/$0.25 No Limit Hold'em Cash, 5 Players
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Holdem Manager - The Ultimate
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BTN: $32.37 (129.5 bb)
SB: $8.47 (33.9 bb)
BB: $25.58 (102.3 bb)
Hero (MP): $27.38 (109.5 bb)
CO: $22.03 (88.1 bb)
Preflop: Hero is MP with J

A
Hero raises to $0.75, 3 folds, BB calls $0.50
Flop: ($1.60) K

4

2
(2 players)
BB checks,
Hero bets $1, BB calls $1
Turn: ($3.60) T
(2 players)
BB checks,
Hero bets $2.50, BB calls $2.50
River: ($8.60) J
(2 players)
BB bets $4,
Hero raises to $13,
BB folds
With that line, I think he has weak Kx, as I think he'd check/call river catching 2pair, especially with no flush possible.
Is the raise OK?
If he'd check the river I was planning 3rd barrel, expecting him to fold his pocket pairs and weak kings. Should I bet here?
Is the sizing good?