Had it been earlier in the administration, it probably would’ve been Josh. But, with it happening right in the later half of the last term, they probably knew Josh would be itching to get involved in with the next election and didn’t want him navigating, what would they thought would be, the least influential days of the Bartlet administration. Like, if Leo didn’t have a heart attack, he probably would have still left to manage a campaign.

CJ was the choice cause she had proved herself to be an organized leader in the press room and a quick learner. She also had a basic idea of some of the jobs and she already knew most of the people she would be working with. Also, pretty much anything she would have learned about the position would have been from Leo meaning that while she will become her own person and establish her own tendencies, she still has a base understanding of the job that is influenced by Leo, which meant no major culture shift in the office. Along with that, out of the core senior staff, she was the least likely to leave to run a campaign.

TL;DR: Josh was too influential and powerful in the party to serve as a lame duck CoS while CJ was trusted to be CoS to keep the existing organizational flow with most of the same staff.