I'm not sure what the problem might be. I've never used the airport router... We have noticed that there is some compatibility issues with some routers. But the Wifi module works with most major brands, so I would imagine it should work with the Airport no problem. The only difference between DHCP and assigning your own IP address is when the router assigns a DHCP address you are can be sure there will be no conflicts of duplicate IP addresses. When you assign it an IP manually you run the risk of having duplicate IP addresses on the same network. Lets say your computers IP address is: 192.168.1.2, so you assign the WiFi shield 192.168.1.3, later a friend comes over and hooks up to your Wifi while the wifi shield was off, you then turn it on and chances are your friend is occupying the 192.168.1.3 IP address and now their is a conflict. But if you use a high enough IP address then there should be no problems with this kind of situation.
I hope this is clear?