How to Rename a Database on a SQL Server Amazon RDS Instance?

RDS has recently added a stored procedure to accomplish this:

EXEC rdsadmin.dbo.rds_modify_db_name N'<OldName>', N'<NewName>'

Source: https://docs.aws.amazon.com/AmazonRDS/latest/UserGuide/Appendix.SQLServer.CommonDBATasks.RenamingDB.html