Wow, I'd never fold there.
You have the second nut flush. What are you afraid of? KT?
I think w/ KT or TT or 44 (or even 22) he would have bet turn.
With a flush he would have bet river or would lose ton of value with a Tx or even Kx checking behind.
I see his river raise more as a "bluff" to scare you with the paired board. Maybe he has the A of clubs, or maybe he has Kx (KQ or even more KJ), which might make sense to check the turn.
You have roughly 50/50 odds to call and I think there are enough worse hands in his range to justify a "call and make a note" attitude.
Because of course it makes sense for him to raise flop w/ FD and check turn. But then he would have just owned you this time by putting you on FD as well and by knowing you'd bet the river. But I still think he has Kx and doesn't want to bet, get raised and face a difficult decision.
Again, only my 2c, I'm curious to the see the real analysis too.
If I'm bothering by commenting in your hands let me know, I just want to exchange some thoughts, fish as I might be