Yes, this can definitely be done without messing everything up.
When I have done this, I have treated it as a new project in my local dev environment, manually copied the theme over, imported the database, and added all of the plugins to composer. Then I deploy the new project and change the server’s root directory to the path required by bedrock. I can’t recall how much search and replace was required, but I’m sure it varies based on your theme configuration and plugins.