Pseudo-fields and custom entity properties from additional schema
You may have to override the form submit handler, or at a minimum add you're own submit handler in addition to the default. As long as the submit handler starts with an entity passed through the $form_state, you could add your own handler to fire before the final handler that saves the entity and set your properties there.

While I don't have a ton of experience working with entities and hook_entity_load etc, you may get more milage from extending and overriding the default entity controller for product and overriding the load, save, and delete methods.