The message:
There is a "Double NAT" in the network. This AirPort base station has a private IP address on its Ethernet WAN port. It is connected to a device or network that is using Network Address Translation to provide private IP addresses.
is purely a warning message, which you can safely disregard altogether.
The annoying feature of this configuration is that none of the devices on the 192.168.0.0/24 network will be able to talk to those on the 192.168.1.0/24 network. In other words, right now they represent two distinct subnets. You may keep this configuration, if this is of no concern to you.
Alternatively, you may wish to integrate all of your components into a single subnet. You may obtain this in this way (the order of these operations matters):
1.connect to the Airport, and disable DHCP;
2.unplug the ethernet cable from the WAN side of the Airport, and plug it into the LAN side.
3.open the GUI of the main router, and change the mask of the LAN, from 255.255.255.0 to 255.255.254.0.
4.Turn off everything;
5.turn on the router, count to five, turn on all of the devices in whichever order.
By doing this, you will have established a twice-as-large network, with these properties (output of ipcalc):
# ipcalc 192.168.0.0/23 Address: 192.168.0.0 11000000.10101000.0000000 0.00000000 Netmask: 255.255.254.0 = 23 11111111.11111111.1111111 0.00000000 Wildcard: 0.0.1.255 00000000.00000000.0000000 1.11111111 Network: 192.168.0.0/23 11000000.10101000.0000000 0.00000000 HostMin: 192.168.0.1 11000000.10101000.0000000 0.00000001 HostMax: 192.168.1.254 11000000.10101000.0000000 1.11111110 Broadcast: 192.168.1.255 11000000.10101000.0000000 1.11111111 Hosts/Net: 510 Class C, Private Internet
This way you will have defeated the dearth of IP addresses, while still keeping all of your devices into the same subnet. Turning everything off and on again forces the introduction of the new mask, and the new routing table.
This works provided you do not have static IPs. If instead you do, you will have to manually adjust the mask to reflect its new value, on each device.