Unless we've got a read that villain likes to play through multiple bricks (the king and a paired card in the hole), chances are he's made his seven-low. If we bet this, we're certainly representing that we can beat one. Unless he really does have a pair to go with his king, I don't think we're inducing a fold here.
I think the biggest positive we could expect from betting here is that we might buy a cheap showdown on seventh, which would be OK if we had a marginal made hand (say a rough seven).
As it is though, a cheap showdown isn't really going to help us. We've got ten outs to improve to a hand that should be a winner (a smooth seven or better) so we'll likely have a pretty good idea of where we stand. If we make something marginal, like a smooth eight, we might want to call a bet from the villain so it'll have been good to have saved the bet on sixth.
I think I check sixth, hoping for a villain check but calling if he bets, then reassess based on what we catch on seventh.