23/20 only 30 hands ...

[converted_hand][hand_history]IPoker, $0.25/$0.50 No Limit Hold'em Cash, 5 Players
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Hero (SB): $50 (100 bb)
BB: $78.95 (157.9 bb)
MP: $38.12 (76.2 bb)
CO: $50.68 (101.4 bb)
BTN: $52.62 (105.2 bb)

Preflop: Hero is SB with K K
3 folds, Hero raises to $1.50, BB calls $1

Flop: ($3) 9 6 3 (2 players)
Hero checks, BB bets $1.43, Hero raises to $4.86, BB calls $3.43

Turn: ($12.72) 4 (2 players)
Hero bets $9.96, BB raises to $72.59 and is all-in

I won't say I know for sure, but it seems to me your x/r size is far too small. The board is not nice, pretty bad for your range and good for BB. They need to call $3.43 in a $9.29 pot, they have position... their decision is easy with 52, 57, diamonds, 7T, 54 etc etc etc : call always. They can also easily flat 99, 96 and stack you on the turn.

I imagine x/c is an OK-ish option here, or x/r full pot maybe. If you take the agressive route you need to grow the size of the pot as fast as possible to deny them their positional advantage. For example if you x/r very big they might feel they have to stack off with a combo draw, because they don't want to call, miss the turn, and now you bet very big (possibly all-in) and they have to give up tons of equity. So they raise you all in and you take the flip, which is not too bad. It's better than letting them see the turn for cheap with still quite some money behind, where they can still call your turn bet profitably due to implied odds.

(And I would snap call turn, BvB I don't think we can fold an overpair)