Worth noting that the CSX - HAK sector would be a domestic sector, and therefore you and maybe also your luggage need to enter China.

This in itself is probably not a significant impediment as the journey is compliant for TWOV (Transit without Visa) in both directions providing that the travel document you have is permitted for TWOV. There's a specific thread in the China forum if you have any queries about TWOV

China Transit Without Visa (TWOV), 2025 onwards

Note the TWOV scheme has had major changes especially in early 2025 and there's out of date info out there on the interwebs.