This trick also applies to the AT&T Uverse routers that do not enable to control DHCP by their users.
There is no option to disable the DHCP in the Airport admin tool, so what you need to do is trick it into being "more passive".
- Set DHCP Beginning Address and DHCP Ending Address to the same value: the IP address of your HDA
- Create a DHCP Reservation (static IP): enter your HDA MAC address and IP address
- NOTE: If the Airport Administration tool forces you to have at least two IP in the DHCP address pool (i.e. if, when entering the same IP address in Ending Address, it adds one to it), it should work fine two. You'll just need to create a second DHCP reservation, and enter any fake MAC address (aa.bb.cc.dd.ee.ff for example).
This will basically allow the DHCP server to only give out one IP address, and it will only give it to your HDA (if your HDA is ever configured to use DHCP, which it shouldn't). All other DHCP requests will be ignored, allowing your HDA DHCP server to be used for all computers on your local network.
BTW, it's much easier if you don't want NAT (i.e. if you have another router that does that), then you can disable both NAT and DHCP by choosing bridge mode. You just can't disable DHCP alone.