Yeah that's pretty gross. I think you played it fine though. River is obviously far far better for villains range. I'm guessing since you 3bet this you are playing a merged pure 3bet strat from the SB so I guess you'll have a tiny bit of 7x (A7s, 87s, (maybe 79s??)) But check 100% on the river has to be right. Then when he shoves you obviously have to call some hands without a 7 but I wouldnt choose this one since it so heavily blocks his broadway FD combos that end up here. I'd choose hands like A8s 88, 99, TT first and Im guessing that's more than enough.

Side note: I think you should be making it a little bigger pre and the turn bet seems worth discussing to me. Do you really value bet overpairs on this turn? Neither of you have much 4x but he should have a little more I think. Seems close to me but I could be totally wrong.

Although the 3 bet preflop should be bigger would still call the river. If villain is capable of having 7x he is very wide so he can also bluff a ton of hands (much more possible bluff hands then 7x hands) as it is unlikely that the SB has a 7 as the SB 3 betted. 78, 79 e.g. should have folded on the flop unless they are spades so that leaves only 67s and 7xs of spades as a reasonable possibilities.
All overpairs and some A2-A5 could also continue as can any Axs

Think it is an easy call on the river.

Yeah the river is probably a call.. would be important to know how big is the rake at your site.

Anyway I would like to ask you if we would check the turn, (not with this exact hand, but in general with some range) would you expect somewhat balanced stabbing range from villain or some exploit numbers?

I would bluff this river, and for me it doesn't matter which hand i have here. Most of the people will fold very often at this type of boards