I've been trying to migrate an entire instance, with all projects and repositories, of a bitbucket cloud instance to another. From what I've seen this is only possible through a script from the API. Is there another way? Can it be done with the Bitbucket Cloud Migration tool?
No. The migration assistant is heavily coupled with Bitbucket Server at the moment. Scripts would be the way to do it.
You might investigate doing a repo transfer: https://support.atlassian.com/bitbucket-cloud/docs/transfer-repository-ownership/
While this cannot be done in bulk, it could potentially preserve most of the data. Definitely test it out first.