Yep, usernames aren't important in LAN racing. Just start a LAN server then get him to connect to your IP. It should work fine, I have two PCs here and I can get two cars on the track, both in demo mode and S2.
Note that unless that PC you sold your friend is either still in your house, or you locked it to demo, you will be in breach of the license agreement.
If you're having problems make sure you can at least ping one another on the network to make sure it is working, but also note many popular firewalls like to block echo replies for some reason.