I need it to be ... on a separate network from the existing one
The new PC will be connected to the server (My PC) through a wireless router
These are mutually exclusive!
If you want the shopfloor PC to connect to your PC (server), they must be on the same network.
If they are NOT on the same network you would need a router in place to connect the 2 networks.
Assuming you have a simple Cable or ADSL internet connection with a single wifi router providing the internet connection and all PCs connected to the wifi router, then all connected PCs will be on the same network, whether connected by wire or wireless.
You can probably limit internet access for the shopfloor PC by using the router setup program to block access for that specific IP address (or only allow access for the other 2, whichever way it is implemented). That works best with fixed IP addresses, rather than DHCP, and depends on the functionality of your router.
One thing occurs to me that might be worth trying:
Using fixed IP addresses, give the new PC a fixed IP address on a different network - e.g. 10.10.10.2, where the existing network is (say) 192.168.0.0 - and give the server network card a second fixed address of 10.10.10.1, in addition to its existing address (cannot be DHCP).
Now the two PCs will be able to "see" each other on the 10.10.10.0 network, but because the new PC is on a different network to the internet gateway, it cannot possibly reach the internet (neither can it reach the second existing PC).
You might need to tweak this by adding a static route on the server, but I don't think it would be necessary.
Having them all on the same network and using the wifi router to block internet access is probably much easier, though!
SD