1/2 NLH Borgata, full table
This is a hand I played a while back and though some of the specifics I might have forgotten, the fundamentals I still remember and thoughts on the play here would be greatly appreciated.

I have around $500 and villain has me well covered: somewhere in the $800 region. He limps from mid position after a few folds before him. I raise the button $12 (standard) w AT. He calls OOP, everyone else has folded.

Flop: K42. Pot: $27

He checks. I have the nuts, I check.
(I admit that betting here is perhaps a better play- sometimes, but as played, my big dilemma comes on the river)

Turn: K422

Villain leads into me $25. I call.

Pot: $77

River: K4229

Villain checks. I bet $75. Villain check-raises to $250. Pot: $407- I have to call an additional $175. I fold.

Am I ever supposed to check back this river. Am I essentially playing my hand face up when I call the turn and bet the river? What do I expect to call me? Any 2X? A smaller flush? He'd have to put me on the Jh if he was trying to get value with the Qh, but Ah is very much in my range. Am I EVER good here- is this a call? Is this ever a bluff? I understand that I could have bet the flop to gain more information, but as played, is checking back the river ever the right play?

I'll add two more things, before the hand there was some conversation where I was bragging about having folded pocket kings before the flop correctly, and he was letting the table know that he usually runs good. When he check/ raised the river he said it again, "I told you I run good".

Also, when he led the turn, I think I detected some acting. He had this bored face on like, "oh well, I guess I'll bet here"- but it seemed a little over the top to me, like he had just filled up but was just pretending to be non-chalant.

Thoughts? Observations? Suggestions?
Thanks !

Puke, this is a pretty crappy spot. I don't mind checking back flop being HU. You flop the nuts, but IMO it's going to be hard to get 3 streets in this scenario. I'd use a mix of c-betting and checking this flop depending on V, and what I believe my image looks like.

As played, I'd fold river. Your hand looks like exactly what it is, and you need roughly 30% equity for this to be an EV neutral call. I just don't think a random V is turning a smaller flush, Kx, or the Qh into a bluff that often. I'd just let it go.

To your original question, I'm definitely value betting this river, and it'd mostly be a bet/fold against non-maniac players. I'd size way down though. It's a 4-flush and paired board, so any hands you're looking to get value from are going to have a hard time calling a large bet. With the pot being $77, I'd bet something like $25-$30.

When you pot the river and get raised on a paired flush board, they have a boat. Bet the flop though.

The flop check is really bad. You have a huge SPR with the nuts and there are a ton of runouts which make it more difficult to get value, like the one you got.

You should definitely bet the river, but should bet smaller. You're asking for a lot from weak hearts.

In general don't bet so much in spots where one card makes the ~nuts and both players can have it. 1/3 pot is fine.