In theory, what you want to do should be possible. You would connect the laptop and PS2 by "crossover" network cable and enable "Internet Connection Sharing" on the laptop's actual internet connection, i.e. the WiFi connection to the router. This would give the laptop's Local Area Connection the fixed IP address 192.168.0.1 and the PS2 would gets its IP address automatically by DHCP from the Laptop.
Now, in order to avoid a conflict of IP addresses, the router and the addresses it allocates by DHCP must be in a different subnet (IP address range) from the 192.168.0.0 subnet used by Internet Connection Sharing.
<<< Alternatively I also have a USB network adapter (for use with a PC in my network) - can this be used? >>>
If that can be installed in the PS2 (and I know nothing about the PS2), so that it can connect directly to the wireless router, that would be a very much better solution.