Synopsis: Network A needs to be able to use a static default route to Network B via a local connection (G0/0/3). Network A does NOT want to route ANY Network B addresses through the rest of its network (currently a number of subnets are being received by other routers inside Network A).
Option 1: Use separate OSPF processes. on Network A's router form an OSPF adjacency with Network B using OSPF process 1. And then form all other OSPF adjacencies within Network A using OSPF process 2.
Option 2: Set the default route on the Network A router to Network B on G0/0/3, then put static routes for the rest of Network A via G0/0/0. This could be a minimal list of subnets since Network A is in big blocks typically. CON: requires human intervention and coordination.
Option 3: Use OSPF inbound filtering using route maps with a distribute list. We can use a route map to prevent OSPF routes from being added to the routing table by matching on the routes coming into gi0/0/3. What I don't know about this is if they'll still be advertised to other routers within Network A. Would the distribute list prevent the redistribution of Network B's routes?
Caveat: Network A admins have no control over Network B's routers. This is why all above options include solutions applied to Network A's router only.