I like the line you took, including the river call.
All streets are standard up to the river where I think we're going to have a hard job finding a fold. Roughly 2.5:1, and if he's got an actual functioning brain this is a really good spot to bluff (I don't like the sizing though).
If he's not up to bluffing
here then there's a good chance he's bad enough to bet AK/AQ which together I think make this a call. I mean, what clubs can he have? The A, K and Q are out there, so he has to have either a hand with a J or T which are significantly less likely, or he had to call your 3bet with a SCer which will happen quite a bit (despite being a horrible play) the chances are slim. E.g. AJ=16combinations, card removal=12combos, combos with Jc in them=3, same with T. + both the J and T are less likely to make the pre-flop call than they are to fold (so the 3 combos are really less than 3 combos..).
So anyway >6 combinations for big flush, SCers are >12 (being really likely to be folded pre-flop), so (significantly) less than 18 all up.
AK/AQ= >12 (since there's a good chance he checks them on the river).
With 2.5:1 to make up for all things possibly wrong with my thought process here, I'm calling. If it was close I'd fold purely because of the amount of guesswork required but it looks like a clear call.