NL25 JJ - thin v-bet on the river in 3-bet pot?

NL Holdem 0.25(BB)
BTN ($37.19) [VPIP: 12.5% | PFR: 0% | AGG: 66.7% | Hands: 8]
SB ($27.51) [VPIP: 33.3% | PFR: 33.3% | AGG: 0% | Flop Agg: 0% | Turn Agg: 0% | River Agg: 0% | 3Bet: 0% | 4Bet: 0% | Hands: 3]
HERO ($83.99)
UTG ($79.05) [VPIP: 35.3% | PFR: 23.5% | AGG: 22.2% | Hands: 18]
HJ ($63.87) [VPIP: 37.5% | PFR: 37.5% | AGG: 100% | Hands: 8]
CO ($27.16) [VPIP: 25% | PFR: 25% | AGG: 100% | Hands: 12]

Dealt to Hero: J♦ J♣

UTG Folds, HJ Folds, CO Folds, BTN Folds, SB Raises To $0.75, HERO Raises To $2.25, SB Calls $1.50

Hero SPR on Flop: [5.61 effective]
Flop ($4.50): T♥ 8♠ 2♠
SB Checks, HERO Bets $3 (Rem. Stack: $78.74), SB Calls $3 (Rem. Stack: $22.26)

Turn ($10.50): 2♦
SB Checks, HERO Bets $5.25 (Rem. Stack: $73.49), SB Calls $5.25 (Rem. Stack: $17.01)

River ($21): K♣
SB Checks, HERO ? ? ?

No info on Villain. Since it is BvB, the plan until turn was to bet, bet/call or jam river. The K on the river kind of stopped me, but he may have some worse hands that could call me. Not having J♠ unfortunately makes him more likely to have busted draws (so I would have certainly called the river donk), but can we find enough of value here?

I assume 2 combos of AK, 2 combos of TT, plenty of Tx. If we can find at least some 8x or 99, especially that all of the draws have busted and K is perceived as a scare card - then we are good to go. What do you think?

Board
[T♥ 8♠ 2♠ 2♦ K♣]

Equity Win Tie
MP2 51.61% 51.61% 0.00% JdJc
MP3 48.39% 48.39% 0.00% 99, 22, ThTc, TsTc, ATs, A2s, KTs, K8s, QTs, JTs, T9s, KsQs, KsJs, Ks9s, Qs8s, Js8s, Ts8s, Ks7s, 8s7s, Ks6s, AdKh, AdKs