|What would need to change is a layer beyond you. Your ISP. Ideally they would do the routing/dns from the 10x net to the 172.x net for you.|
No amount of cabling or your equipment can change that.
Your router should have the ability to save and restore system configurations.
Make one for games, save it.
Make one for 'net, save it.
Then when you want to play games you load the game profile. When you want to get on the internet, load that profile.
Or get a cisco router and put subinterfaces on the wan port for each subnet and the router will take care of it for you.