Assuming the gateways are on different subnets, just assign an IP address from each subnet to each card. If each subnet has it's own DHCP server, you shouldn't even need to assign addresses to the cards as they will pick them up themselves.
I guess, the separate networks for applications and internet are designed like that by a system / network administrator? In this case it's not a good idea to connect to both networks with one machine. At least you should switch of the 'internet connection sharing' in windows. Otherwise the internal network is reachable from the internet trough your machine.