|"All the PCs on your local network will share that same IP address." --> so this is not true.|
Oh but it is!. All PCs on the same network share the same external or public IP address. That is the one the rest of the world sees regardless of what IP address is allocated to each individual computer which will be an internal private address.
What you are trying to do will not work as there is no such thing as a virtual IP Address. It could be made to work if you you install a second Netwrok Interface Card and allocates that a real IP address in the same range as the rest of the network. A NIC can only work to one IP address at a time.
However, it still wont do anything for your download restrictions. That is dependent on your public IP address and that is allocated by your ISP.