Checking your option in the BB seems correct to me here since a correct raise size would need to be large and we then would be bloating the pot OOP with a very marginal hand. Checking behind is safer and makes life easier (which I know we should not always be trying to do but in this case it makes sense).
I would prefer leading the on the flop here since it is a limped pot and we have top pair. I would much rather try to take control of this pot instead of sitting back and letting someone else dictate the action. Much easier to pot control if we take the lead even if we are out of position. As played I would also call behind as you did as I believe their range would be Ax and Kx heavy. Many Ax have us and if we check raise I believe a Kx folds and we lose value.
On the turn I think you have tougher decision here then you might have thought at the time (I'm just guessing on that). I don't think a Kx that isnt two pair would double barrel here so I would remove that from their range and strictly put them on Ax. All Ax beat us now except A2, A3, A5, and A6. Thats a pretty small range of cards and I would highly consider folding. I am not sure in the moment I would have folded but there really isnt much we beat given villains actions.
On the river its definitely a clear fold to me now that one hand (A2) from the turn range that we beat is now beating us and villain triple barreled with sizing that was consistent at a half pot each street. We only beat A3, A5, and A6 now and unless you have some info on him that he is capable of triple barrel bluffing then its a clear fold.
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