Store information from hook_form_alter to be used in later hook_user_update?
As far as I'm aware, all form element values entered by the user on the profile edit form are available in the hook_user_update()'s $edit argument. The only element that won't be there is the plain text password.

There is no "drupal way" to avoid this, as core handles this functionality and it's trying to stop you from accessing the plain text password.

REMOVING SUGGESTION PER @znerol. Will update later this evening