b/f turn best line?

PokerStars No-Limit Hold'em, $0.50 BB (7 handed) Poker-Stars Converter Tool from FlopTurnRiver.com (Format: FlopTurnRiver)

MP2 ($12)
CO ($61.15)
Button ($54.95)
SB ($64.40)
BB ($66.90)
UTG ($54.70)
Hero ($78.10)

Preflop: Hero is MP1 with K, Q.
1 fold, Hero raises to $2, 2 folds, Button calls $2, SB calls $1.75, 1 fold.

Flop: ($6.50) K, J, K (3 players)
SB checks, Hero bets $4.5, Button calls $4.50, SB folds.

Turn: ($15.50) T (2 players)
Hero bets $9.5, Button raises to $48.45, Hero folds.

Final Pot: $34.50

Buttons shove was very fast, like 3 seconds after my bet. Only saw one significant hand where he had KJs and called down on K high flop and lost to AK. Seems pretty donkish, no stats or other reads. Another boring hand...

Barring reads, I like a check/call here unless he over-bets the pot. You have a lot of outs against the flush and he'll check non-flush hands behind on the river.

rationale behind betting turn here please

idk I guess for value? But that sure is one sick turn card and probably everything I beat folds and everything I'm behind to raises (or calls to slow play ldo), right? Soooo I should be c/c a decent bet, and c/f a large one? If I c/c and boat up shove river? I hate folding so much....

There are no hands that you do beat except for a pair

i think i marginally prefer a c/c but its a pretty gay spot in general