I don't know much about RDS, but as I understand it is just SQL Server. It's not like Azure SQL Database, something special. This means that you could just take a backup of the database (WITH COMPRESSION!) and download and restore it. Although that presumes that Amazon let's you take backups. And that you can access the backup.
Now, I am here thinking that you can take down the application for the entire period, and I will not be surprised if you tell me that this is out of the question. If take full backups combined with COPY_ONLY log backups, you may be able to cut down the downtime.