Yes, every router I've seen will let you chose. Some manufacturers have different terminology.
For example, in some routers, turning off 'gateway mode' accomplishes the same thing as turning off NAT, etc.
Keeping the firewall enabled will make it a little trickier, but can be done.
Assuming you're not going to need a lot of bandwidth between the networks, then your basic Linksys/Netgear/DLink etc will work. If you need something with more horsepower look at their business-class routers.
Once the router is in place, the only other thing you'll need to do is to add a static route in each of the existing internet routers pointing to RouterC for the other network.