Restore specific MSSQL backup from media set using AWS rds_restore_database
Seems from the documentation that the procedure doesn't support specifying which file within a .bak you're actually trying to restore. The native RESTORE DATABASE command allows you to specify this using FILE = .
I think the solution in your case is to always back up any database to a single .bak file (name the backup file uniquely, and always use WITH INIT when backing up). Combining multiple backups into a single file doesn't have a single benefit that I can think of, especially combining multiple backups of different databases.
To get to the backup you want, you may have to download the multi-file backup file, restore it on your own instance using RESTORE DATABASE ... WITH FILE =, back that up to a new, single-backup .bak file, and upload it back to AWS. But I would consult with AWS support in case they have any other (easier) options.