Hey Marlon,

I assumed that we're facing a relatively tight, somewhat straightforward opponent, whom I believe to be the standard PLO200 opponent.

His range :

I assumed that Vilain would have a tight stabbing range on this flop, and given that he probably would have 3bet the vast majority of his AA, and that 66/33 should rarely be in his range and on top of that you're blocking a 6, we shouldn't expect a lot of set combos.

If he's stabbing a strong range, then I believe that his range should contain a few combos of sets, a few 63dd (or some other draw), good straight draws and combo draws, a lot of A6/A3 and A:dd which represents at least 35% of his stabbing range in my opinion (I messed a bit w/ Odds Oracle but I'm not an expert)

When he calls your turn x/r, my guess is that we can basically remove 54 and most naked two pair combos from his range. Some people will call here with say, A6KT!dd, but you look so nutted on this turn that I believe most opponents would fold those hand with so few clean outs.

He's likely to shove sets+fd, and might shove a set+straight draw or even a naked set a small percentage of the time. It's very player dependent at those stakes so I really don't want to make any bold assumption. But add that to the rest and I think that he should have very few sets here.

All in all, I believe that once he called your turn x/r, his range is heavily weighted towards A:dd, with a minor proportion of A7/A6/A3+redraw for some and combo draws, and that means that his range will be very bluffcatcher-heavy by the river, have a few air hands and a small percentage of full houses.

When the 7 pairs the river, it's quite unlikely that he's got a boat. He shouldn't have a lot of A7 in his range, very rarely 33/66 and 77 in an insignificant proportion.

Kind of a wild guess, but I believe his bluffcatcher/valueshove ratio on the river to be over 4:1. I don't remember the exact ranges that I put in in Odds Oracle but I ended up w a number around this.

Edit : Actually I'm probably wrong about this and I believe that it might be higher than this. 

Your range :

On the river I can only guess what your range looks like since you didn't give info on your turn x/raising tendencies. 

My uneducated guess is that you're quite heavily unbalanced towards nuts and have strong two pairs+strong redraws and combo draws in minor proportions.I'd guess that you probably have a value/bluff ration between 3:1 and 4:1 but this could be off.

On the river you got 0.63 PSB left, therefore if you're shoving you need to be bluffing 27.8% of the time for him to be indifferent to call with his bluffcatchers.

If you want to shove all of your bluffs, you probably need to shove almost all of your straights and your boats whenever you got one, which should be pretty rare.

In a vacuum you probably can x/c your straights profitably on the river tho, because he might have some busted-draws that he wants to bet, but I don't think that it's the best way to play it.
I believe that you'd be missing on quite a lot of value with your whole range, and exploitatively you look so strong here that I don't think that the average opponent would feel like shoving his busted draws here(Even if he should do it at some frequency).

Exploitatively speaking I think It would make a lot more sense  to x/c if the hand would have gone flop checked around, your x the turn, he stabs the turn and you x/r and he calls because he'd be much more air-heavy on this river because his range would consist of more busted draws and less bluffcatchers, and even if I don't think that he'd be very likely to bluffshove, well, he'd have a lot more opportunities to do so.

In the end I believe that your range is much stronger than his range on the river, and that his range is mostly bluffcatchers and a few very strong hands.

In this situation I believe that the best line is to simply shove your whole range. By doing so you're targeting the vast bluffcatching part of his range and you're making a very profitable bluff with your busted draws.
Basically when you shove you don't allow him to checkback his showdownable hand for free and you put him in a spot where most of his range has what I believe to be something close to a neutral decision.

Concerning your idea to x/f,

I don't think that he's got enough boats for you to be very worried about it to the point where you should consider x/f anything.
54 is probably the top of your range on this river, except maybe for a rare A7dd or 77, and a few 66/33/AA combos that you decided to slowplay on the flop.
Therefore you would need a very strong read to fold the top of your range against a shove. You would need to know for sure that he would almost only reopen the betting with a full house. Even If I don't think that he'd bluffshove a lot on this river, I wouldn't x/f my whole range without a super strong read.

I'd rank your options like this, and I don't think it's very close :

shove

x/c

x/f

If anybody else have some critics on this post I'd be happy to hear them, there's a lot of assumptions in this post and some of them could be off and affect the whole logic behind it.

hope you find this helpful,

Matt