I've never configured HP switches (just Cisco) but based on the specs, yes, you can do what the customer wants. The installer may not know how to configure the switch which is why he probably wrote it off.
You can create a VLAN on the switch just for the ports the cameras are plugged into. No need to use the dual NIC's. Just plug the computers into the same switch. If the customer wants to have the computers on the same network as the cameras, just make sure the ports you plug them into are assigned to the same VLAN. If they want the computers on the other network, you can assign access lists to allow just those two computers to talk to devices on the VLAN. Just make sure they are all plugged into the same switch.