So you are connected to the airport with wireless?
You can run dual networks.. here is how..
The main internet connection and the secondary connection must use different connection types.. so one is ethernet and one wireless. The main connection is with the internet link, so you need to keep it using router.. ie gateway address and dns.. the secondary connection must not have those.. it is simply IP and subnet mask.
So to get this running you still need to isolate the computer from the internet for the time being..
Before doing so open the network preferences and find the current IP of the airport. ie by default it is 10.0.1.1
So to fix up the TC you have to move its IP address. Make it 192.168.1.1 for example.
Doesn't matter what you use just it has to be different.
Then setup the local network using static IP. So for ethernet.. 192.168.1.10 and subnet mask 255.255.255.0
You don't need anything else.
Now allow the computer to connect to the internet and the TC at the same time.. open the airport utility and make sure you can still access the TC.
If you run into trouble.. give me screen shots of what you setup.. I should have asked you to do that already.. perhaps something is set wrongly and it is much easier to see in a shot.
Don't get frustrated.. take a break.. if some of this stuff is new it takes a while to absorb.
I need to know what your computer connection is now to the airport.. what IP it is on.. ? Then I can be more specific.
Does the MacPro have dual ethernet??