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i'd shove. if you can't get paid off by worse here you should bluff more. i think he's a 1 pair or 2 pair hand way way way more often than he's a flush.

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2pair is possible, but I think 2pair would have reraised turn once it was HU b/c of the draws out there, or if he didn't raise turn, now fold the river b/c the flush came. Both of these scenarios would advocate checking.

Plus remember how strong HEK looks here to villian.
He raised the turn into 3 opponents (an important observation here I think), and is now betting again when the backdoor flush hits. I think villian is unlikely to call 2pair or lower here.

So unless read on villian is that he will call light or that HEK has a laggy image, neither of which he mentioned, I still believe checking is the stronger play.