Finally got this working, so I'm posting here in case anyone else has issues:
I started the SMTP service under IIS on a local server (I just used the sql server machine for testing). Then, I configured an SMTP virtual server, "[SMTP Virtual Server #1]" with the following properties:
Access | Authentication : Anonymous Access
Access | Relay restrictions : Only the sql server machine (optional)
Delivery | Outbound Security : Basic Authentication (username, password of external smtp)
Delivery | Outbound connections : TCP port: (whatever port external smtp uses)
Delivery | Advanced: Smart host: (address of external smtp)
Then, in the Reporting Services Configuration Manager under E-mail Settings, enter the LOCAL IP ADDRESS of the virtual SMTP you just configured and the sender as a valid user on your external SMTP.
One side benefit: this should also enable the crippled SendMail task in SSIS if you set the local ip address as your smtp server in the smtp connection for the package.