Make sure the document's password security compatibility is set to "Acrobat 5.x or earlier". If there is a Master password in addition to the User (read-only) password, put the Master password in the Stamping URL; otherwise put the User password in the URL if the file has no Master password.
If you can get the Stamper to work, but it doesn't seem to retain the original User password, try adding a Master password which is different from the User password, put that Master password in the Stamping URL, then see if the User password is retained in that case. If not, then that's just the way it has to be with your particular file. Note that the Stamper always adds a random Master password to the file anyway, which prevents any extraction or other tampering with the document's content and settings.
Bear in mind that even using a User password in this scenario, where you are giving out the password along with the file, effectively does nothing to improve the file's actual security nor discourage sharing, since anyone who wants to share the file would simply include the password. It's like locking something with a padlock but then leaving the key right there in the lock, so what's the point of locking it at all if everyone has the key?