Princess Katherine was not truly trying to help Kang Chen; she was acutely aware that both she and Kang Chen were in the same boat.
If Kang Chen truly resigned, then the target of public attack would shift to her.
Kang Chen is currently the one bearing the brunt in front of everyone, his resignation seems to be his downfall, but in reality, isn't it also a means of preservation?
On the contrary, their crisis would truly begin once Kang Chen steps down.