Can my client prefer not to say?
There cannot be a prefer not to say when the information is being obtained for identification purposes. "Sex" is a required field so this cannot be optional. The sex field "Male" and "Female" options are the identifiers used when running the necessary electronic checks.
We are not aware of passports having an X on them as gender is always male or female under HMPO rules. When someone transitions, they can apply to change their passport to the other gender.