KK river sizing logic check

PokerStars No-Limit Hold'em, $0.10 BB (5 handed) - PokerStars Converter Tool from Play Online Poker, Site Reviews & Poker Forum | FlopTurnRiver.com

SB ($8.69)
Hero (BB) ($17.68)
UTG ($11.55)
MP ($20.43)
Button ($19.69)

Preflop: Hero is BB with K, K
1 fold, MP bets $0.30, 2 folds, Hero raises to $0.90, MP calls $0.60

Flop: ($1.85) 7, 6, 3 (2 players)
Hero bets $1.20, MP calls $1.20

Turn: ($4.25) 7 (2 players)
Hero bets $3, MP calls $3

River: ($10.25) A (2 players)
Hero bets $3, 1 fold

Total pot: $10.25 | Rake: $0.46


My logic in betting small on the river is because i could get called by worse 88-QQ and i didn't want to bet say $7 and get jammed on as the river was a bad card for me. Although i hold blockers to AK a hand he might check call 2 streets with there's still a few combos out there and he could possibly hold AQ as well.

Although i felt during play he was more weighted towards TT-QQ and AK sometimes as well but not as much because he seemed to be playing pretty snug.

Thoughts?