I went to the Apple Store to ask this question last year. I wanted to save all the SMS's between my partner and myself as a keepsake and because re-reading them all shows how our relationship has developed (for the better).
The Genius told me that there are Apps available that will export your conversations for you. However, these are not created by Apple and they require many permissions to access files in your phone. I looked into it, but I wasn't brave enough to get one. You can search for "Text to Email", or "Print Text Messages" and check them out.
I suppose that privacy won't matter for you in this case, because you can delete the App when you are finished, or it will be deleted anyway when you factory reset.