Unfortunately, there are 4 cards to a straight on the board, on top of the flush potential, so we're far more likely than usual to only get action when we're beat. There are also a lot more ways to make the hands that beat us (an 8 or any 2 clubs) than the hands that we beat that could conceivably call (all require 2 specific ranks for two pair or a set). She'll probably also fold one pair or less, so nothing is gained by betting in any of those cases. I doubt we can bluff her off a straight either.
Maybe, against some opponents with whom I have a narrow read, I'll go for value here, but Villain's range is wide open—especially since we didn't bet the turn. She probably doesn't have a flush, but she could easily have that one-card straight and is checking out of fear of the flush (or slyly value-checking it).
Check behind, be grateful for the free showdown. If we lose some value to a weirdly played set, so be it.