How to translate content of the composite element settings?
You need to create a schema config file to make them translatable.
I have done it like this by creating my_module.plugin.handler.schema.yml file in my my_module/config/schema directory.
webform.handler.my_element:
  type: mapping
  label: Booking email
  mapping:
    booking_email:
      label: 'Something'
      type: text