You should be able to connect to both networks on the laptop simultaneously. You do not need to choose to connect to one or the other. Each PC will show the other system once this is configured properly within the Network section of File Explorer, but the network adapters themselves will not show up there (only within the Network and Sharing Center). I'm trying to find a suitable tutorial for setting up network file transfers rather than re-writing the whole thing out here (it would be quite a long post if I tried to account for all potential problems).
Also keep in mind that the transfer speed over the wired network via that wrt54gs will be capped at 100Mbps which is WAY slower than a flash drive. Assuming unobstructed transfer rates between the two machines, a 50GB file will take over an hour to move. Via a flash drive (even a USB 2.0 flash drive), the time required is cut to around 25-30 minutes, as the maximum transfer rate for them is 480Mbps. Via a switch or router with gigabit ports, the time is cut further, to somewhere between 6.5 - 8 minutes (assuming the gigabit line is fully saturated, which won't happen unless you bought a new hard drive in the last 6 months, or are using a solid state drive). Just something to keep in mind if you plan to move this file around a lot.