BTN checked back on the turn showing some weakness. Will still have some over pairs in his range, but that is a very small part. I would size smaller on the river around half pot.

personally I think that on the flop it should either be a lead or a checkraise. the range advantage is such that a lead on the flop would be easily +ev, leading is a strategy underused by players. checkraising is a valid option as well, we are out of position and want to take control of the pot, both players are deepstacked enough for this to work. People tend to overfold to checkraises so this, combined by all turns, apart from A/K are favourable for our range, including all diamonds which creates a backdoor flush draw.

Personally, I would lead 40%, if he calls, bet 75% or 130% on turn, we have a massive range advantage and cash easily push our edges. He is likely to fold over pairs to our turn bet, should he call, we can give up on river, or empty the clip, depending on the draws that missed or got there.
With the specific run out of this hand, I would bet 40% OTF, bet 130% on turn and shove river, as he has moreu flush draws that call(and miss) than you do, so you are going to make profit if you take this route.
What do you think?