AK All In on Turn HU

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AK All In on Turn HU

Blinds: $0.25/$0.50 (2 Players) BB: $73.50
SB: $50.11 (Hero)
Preflop ($0.75) Hero is SB with K A
BB raises to $1.50, Hero raises to $4.50, BB calls $3.00
Flop ($9.00) T 4 J
Hero bets $6.00, BB raises to $14.00, Hero calls $8.00
Made a standard c-bet on the flop, when I got raised I kind of assumed a flush draw or top-pairy kind of hands so I called with my two overs and a gutshot.

This might be my first mistake as it's a very small raise to make with a flush draw.
Turn ($37.00) T 4 J K
Hero checks, BB bets $55.00 and is all in, Hero calls $31.61 and is all in
Was planning on check raising all in as I probably have the best hand a lot of the time. I think he he can have hands like AJ, QJ, flush draws which I'm ahead of as well as all his two pair+ hands. When he shoves I'm not entirely sure what to think. I think he can still have his range of Jx, flush draws, (maybe QK?) which I'm beating and his two pair+ hands. As I was planning on getting it all in I decide to call.
River ($123.61) T 4 J K 5

Am I just spewing off in this spot? Do you think that more often than not I've turned the best hand? I've seen him barrel off in a couple of 3bet pots so that also lead to me calling off when he shoves.

This is kind of another range question as I've run the maths on the range I assigned him and I'm pretty much flipping. I've given him top two pair, all sets, QQ+, TPTK, TPSK, pair+straight draw, A high flush draws, some K, Q, J high flush draws, and a few suited connectors (hearts). I'll list it all out at the bottom.

Is anyone playing this differently? Anyone folding the flop/turn?

Assigned range: TT+, 44, AJs+, KTs+, QTs+, JTs, Ah9h, Kh9h, Qh9h, Jh9h, Th9h, Ah8h, Kh8h, Qh8h, 9h8h, Ah7h, Kh7h, Qh7h, 8h7h, Ah6h, Kh6h, Ah5h, Ah3h, Ah2h, AJo+, KTo+, QTo+, JTo

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