I attempted to switch to DHCP/NAT mode, but then I get an error message stating "The DHCP range you have entered conflicts with the WAN IP address of your base station".
If you will reset the AirPort Extreme back to factory default settings, and then set it up again, the AirPort will use the 10.0.1.x IP address range, which will not be in conflict with the 172.16. range provided by the Cisco device.
If it were me, I would power off the Cisco modem and leave it off for 20-30 minutes to allow it to fully reset.
During the time that the modem is powered off, I would reset the AirPort Extreme back to factory default settings, then power it off.
After the modem power down period, power up the modem and let it run for 5 minutes.
Then connect the AirPort Extreme to the modem and power up the Extreme, wait a minute or two, then set up the AirPort Extreme again. Depending on whether the AirPort thinks that the Cisco modem is a modem or router will determine how it will be setup up. If the AirPort winds up in Bridge Mode again, then you can manually change the setting back to DHCP and NAT and the AirPort will use the 10.0.1.x range for IP address assignments.