It's not ideal, but I don't think it's unreasonable for the indexer to need to be configured with the repository URL(s) in order to support reindexing.  Another option would be to have the REST endpoint take full URL(s) instead of just paths relative to the configured repositoryURL.  That would allow maximum flexibility.

With repositories that require auth, I don't think there's any way to get around having to configure the credentials.  But we could also take advantage of them and use all of the configured repository URLs as the default reindexing targets.