Even though its a super dry board with this many players in the hand I would bet 1/3 pot to get out all the weird pp's that would hit a set on the turn and get value from a worse K. The reason why you always bet the flop when you flop a huge hand is the same reason you would do so here.
You played the flop "sneaky" and then x/r the turn which turns your hand face up instead of betting small twice and keeping your hand more disguised. A backdoor flushdraw isn't actually a BDFD when you check the flop, its just a normal FD. Pretend the board was K
K
4
and the turn was a 7
and the V made this bet. It would most likely be a flush and this is effectively what your flop check made this hand into.
V could have 44, 77, or a flush. Its an obvious fold and if he somehow shows up with a sick bluff here you just tip your fedora and say well played.