I think you got lucky...I would hazard a guess that you were unsure where you stood when your last $435 went in on the river?

I think, taking the hand as is rather than a lot of what-ifs at other points, that other than folding the turn you pretty much are committed to playing showdown poker (i.e. whoever shows down the best hand on the river) with no room to 'outplay' the opponent. And you will have to be prepared to play for the remainder of your stack (assuming Villain covers). If you cover Villain, then you should be prepared to play effective to his stack. Either way, the SPRs get a little funky for a one pair hand and not even TPTK at that. The reason is:

A) if you flat call is $150 turn bet, pot becomes $568 for river action. With a brick and V1 jams, then you're getting about 2.3:1 ($1023:$435) for your last $435 so you only have to be right about 30% of the time. While there are 8 combos of AA and 8 combos of AK (while a little less likely with your holding KQ, but not outside a reasonable possibility considering all other action). ABC players here aren't likely to 4-B AKo OOP and only above average aggressive players, even this deep, would elect to 4-B AKs OOP as pot becomes bloated and if they wiff flop they're very likely forced to c/f to a(ny) bet. And with six combos of 88 and 77 each (even though I would discount about 50% of 88 and 77 combos as many/most ABC LLSNL players don't open raise these middling pairs in EP, even though they should in a lot of different types of games). At 1/2 and 1/3 levels I generally observe the majority (but not all) average opponents to play pretty ABC so I think you're right on the equivalent threshold to being ahead/behind.

B) In any instance where the Villain has to turn his hand into a bluff, he would have to be confident you're holding one pair of K's even if it is TPTK to get a bluff shove to work any % of the time; and even if he thought you had AA, you're still calling (same pot odds and hard to fold AA, even here). Also, wouldn't take math genius to figure out that he would be offering you more than 2:1 with a shove on a non-threatening board. If Villain checks river, I would check behind as then the reverse is set up, he's getting 2.3:1 and is he gonna fold a better hand to your river shove? And what worse hands are you expecting to call (other than what he actually held).

C) If you raise Villain on turn, even a min raise, then SPRs just about force to you call regardless of whether Villain calls and jams river, or 3-B jams turn and you are left unsure of the strength of your hand and just playing math odds of SPRs.

I admit, after flatting the donk lead, I probably would've have folded to the turn bet seeing as I would've felt funny playing for my remaining $585 with 'the hand of death' and with no real re-draw and five outs at best. I would prefer to get my 350 BB stack in with greater equity.

The only thing I might have done differently is to raise flop donk lead to $140 and evaluate Villain's reaction. A flat call followed by a check on the turn (which I would check in-turn) gets me to a 'cheaper' showdown if Villain checks river; okay odds if Villian picks a value sized river bet, and a fold to a jam - forfeit $195 and look for a better spot later on against this opponent and/or others.