Personally while this job on the surface looks like an awesome opportunity, I don’t think it is exactly what Kate wants. She’s always been about the personal touch in being a cop and helping the victim’s families work gain closure by solving the crime, even the ordinary, mundane ones. That is reason enough to eventually turn it down. Also, the job sounds like it will take up all her time and even if Castle were to move to DC, she will be giving up much of this newly acquired personal life she’s invested. I think that’s also reason enough to turn it down. But season finales now much end in some cliffhanger (which I think is not necessary) so we probably will be no different on Castle.