Is it a condition for a preacher to acquire the status of "preacher" that he be a complete memorizer of the Holy Quran?

It is not a condition for the one who invites to Allah to be a memorizer of the entire Quran. Rather, he can invite to Allah with what he has learned from it, even if it is only one verse, due to the saying of the Prophet, may Allah bless him and grant him peace: "Convey from me, even if it is one verse."