If you simply set up NAT on the router you are immediately behind, and make sure the 192.168.1.0/24 subnet doesn't exist on your side of it, it will automatically treat that address as an Internet address and do NAT for you. That it is an RFC1918 address is of no consequence.
The packets will appear to come from the NAT host. If that is not 192.168.1.3, you can't do that (the packets' return path would be invalid).
However, depending on your network architecture, you may not need NAT between those subnets. It is perfectly legitimate to use a routing protocol (like RIP) and allow the routers to route between those two subnets. The only implication of using RFC1918 addresses in this case is that you can't traverse the public Internet using them; on your intranet they can be used like any other address, including for subnetting and routing.