Nut loose,
I'd be very, very dubious that much, if any, legacy CH-53 kit/experience will be compatible with -53K. The KS was optimised by the USMC, who's remaining -53s are quite different from the GAF cabs. Yes, it folds into the same space but pretty much everything else is different.
From a Project point of view the CH-47F is a far safer option. The UK, Dutch, Italians & Spanish all fly it, as does the US Army in Germany. There are opportunities for huge Through Life Cost savings by sharing facilities (e.g., the Dutch have used the RAF sims at Benson for years), building the airframes in Europe (through the Finmeccanica licence) and benefiting from joint efforts on TTP development and future upgrades. By comparison, the -53K is high risk, likely to be more expensive and there are no European partners.
The three key factors will likely be the requirements (which seems to confirm that both are suitable), the cost model and which way the French go.