For computers with RFC 1918 IP addresses, they need to have a public IP, in order to communicate to the rest of the Internet. So, the router will need to run dynamic pat with an outside IP address in order for this wo work.
Also, when the return traffic comes back, from the Internet, the router will need to know how to get to the Cisco SF300 which has the routes.
This of course all depends on your entire network topology.
Sounds like you just need dynamic PAT configured for the rest of the vlan interfaces