Hello,
I checked the MXIC comparison of these chips and they are very similar. The main difference with the 35E is the physical pinout is different.
Do you mean you compared the datasheets? I suspect that multi-chip definition MX25L12835F/MX25L12845E/MX25L12865E (which at head already grew to MX25L12833F/MX25L12835F/MX25L12845E/MX25L12865E/MX25L12873F) can cover you chip as well, but this needs a look at the datasheets to tell for sure. That's why I am asking.
I am using a System Rescue USB drive to run Flashrom
Are you rescuing the system which is already broken? If you make a mistake, would you be able to re-run?