I'd definitely bet bigger on the turn, and probably also bet bigger on the river. Sure you don't have that many bluffs, but there is a lot of combos of 89 and also you can at least be perceived to have like J9/96s some of the time i'd guess. Also at these stakes i think just using an exploitable (bigger) size will probably be better... balance isn't always important. The river sizing may be fine, but i think the turn needs to be bigger, i'd go for ~$4.40 on the turn.

I like the bet-fold though. It's annoying, but good discipline to lay it down.

I would call the flop. As played, it's gross but it's a fold.
I would bet bigger on the turn and a little smaller on the river because you want a call from Ax hands because he doesn't have Kx except for AK.

So you raised the absolute top of your range and you get to the river and it's still the top of your range and now you have to fold. I think the situation speaks for itself, you shouldn't raise on this board when he crush yours and it's fairly dry. So it's not like the nuts change on every single turn. I prefer calling and getting a x/r in on later streets, typically the river.

Definitely just calling the flop, as your range is really far behind his.
River is a pretty clear fold. Nobody's jamming trips for value, or as a bluff. That's a pretty advanced play.

I would like to thank everyone for the input. Also I am really glad I posted this hand. The result of this discussion is very interesting, because I never doubted my river fold. I hoped you will preferably depict my line overall, without me pointing the flop out and swaying your comments wrong way. Thanks again guys.

I think your line is fine but that we should x/c the river. My ideal x/r range on flop would contain some AT (not all) and some 77 (not all) and a smattering of gutters with bdfds plus an occasional OESD. A very big thing has happened since the flop though in that on the river villain's range more than doubled the number of combos which are beating this value hand (AK probably has higher flop betting frequency and lower flop 3betting frequency than AA so this really is not a small thing) and our AT value hands are splitting with any ace. When such a catastrophic thing happens to our range we can stop betting so much and start thinking about checking to protect our hands and get them to showdown against whatever it is that they do happen to be beating still (AQ, for example).

I think that the people advocating not having a x/r range here are being silly. We shouldn't x/r often but when we have 77 otf here we are beat by 6 combos out of in the vicinity of 550 combos which BTN opened. We're behind around 1% of BTN's range. We could easily overbet donk three times to get all-in on the river and still be getting clear value against a BTN who tried to defend enough to stop us doing so profitably with 43o. It's completely fine to be x/r'ing this hand sometimes and it is a more effective way of protecting the rest of our range, which overall is somewhat weaker than BTN's, than x/c'ing this hand always, because x/r'ing lets us x/r bluffs as well and because x/r'ing makes villain less inclined to cbet tons of hands for large sizes and a lot of our range is stuff with at least a gutshot or bdfd which would really love it if BTN didn't cbet tons of hands for large sizes.

The third point is that this is actually a board which changes a ton and changes very favorably for us. Any K, Q, T, 9, 8, 7, or 6 is a card which gives our x/c range plenty of strong combos to defend with against BTN's range, which is mostly ahead of ours because it has AAA and TTT, which stop mattering so much due to straights a lot of the time, and AK, which stops mattering so much as more hands make two-pair or our Ts and 7s make trips. So, just because BTN has some better nutted hands than us doesn't mean that he's going to be able to give us all that much trouble on later streets if we're not x/c'ing 777, and even if it did 777 isn't even a combo which helps defend us against AAA and TTT, if those are the problem the hand we need to be x/c'ing with is AT because it blocks them both.