We have a Vigor 2820 router in the school I work in, and the content filtering is done on the line before it enters the school.
However, it is possible to set up different filtering levels based on the external facing IP address.

I know it is possible to set up another internal IP address for the router using the, but is it possible to bind this additional IP address as a gateway with a different public IP address?

It sounds like you want to use Address Mapping which is under the NAT menu on the router - that'll let you specify which IP address is used by specific IP subnets on your LAN that would use the public IPs that you have available that can have different filtering applied to them.