I moved from CA to NC during June 2021 and continued to work for my CA employer for the full year (as a remote worker). In 2021 W2 box 16, my employer reported CA and NC income based on a move date in September 2021, as I think they changed my designation to remote worker at that time
I have spoken to the company payroll team and they will not be correcting the W2, as it is based in info in their system i.e. approval of remote work date. They have asked me to make the appropriate changes in the state tax return and keep relevant calculations, in case the state tax authority have follow up queries.
You can't change the box 16 amounts on your own. Since your employer won't change them, the simplest thing for you to do is just file the way the employer reported it.
The California Franchise Tax Board (FTB) is very aggressive about collecting every cent of income tax that they think you owe. If you report less California income than what your W-2 shows, you will end up having to try to convince the FTB that the W-2 is wrong. Since your employer does not agree, that will be very difficult.