From what I've been told the K can't do the mission. Something about too much carbon fiber and not designed with tow stresses considered.
Rumor mill from friends in the 53 community: the navy had the chance to buy into the Kilo so Sikorsky could add the extra frame stiffening necessary to pull the sled and big navy said "no thanks". So not much hope for the 53K to be pulling a sled anytime soon.