>>> So the laptop is working fine at the moment but
>>> it can't be taken anywhere that has a router on
>>> the 192.168.1.0 subnet (is that the right
Terminology is spot on! As you say, in order to use the Laptop with a router that operates in a different Subnet, you would either have to change back to automatic IP addressing, or give it a new fixed IP address to work with in the new Subnet.
You could try "hard resetting" the router back to factory default settings and setting it up again from scratch, which might sort its DHCP problem. Otherwise it sounds as though it is faulty. There might be updated firmware available on Belkin's website which you could install as another option.