Very good.. now I know what is happening.
When the modem has a connection drop, (it sounds like you have it bridged), the local interface of the modem passes the internal address of the modem.. until the public IP is restored..
So that is why you get this affect..
Using internet > stops working > Airport Utility > see "Double NAT" and d/c'd internet > restart AEBS > wait until it starts up> double NAT goes away and now connected again > resume using internet.
I doubt the double NAT is the cause of the problem .. it is a symptom of a failed connection from the cable supplier.. (Telstra??)
Because the TC picked up the internal IP of the modem it is now stuck with that address until the dhcp lease expires.. hence why you need to reboot the TC to get back to internet.
I would recommend you go back to using the CG3100 which is a router, in full router mode.. and bridge the TC. Keep track of the daily dropouts in the log.. and just keep hassling the ISP to fix the line. Cable has join issues.. and ISP can get lazy if you still have some internet.. dropouts on cable should be rare.. not daily events.
I have also suffered this issue with a previous ISP modem and the 4th gen AEBS. I could never fix the issue for some reason. It would just randomly come on, and then I'd just have to let it right itself. I hoped these new models would sort it out.
This fact matches the above.. the cause of the problem is not the equipment.. it is the line you are given.
If you actually access the modem interface.. typically 192.168.100.1 or 192.168.1.1 you might be able to turn off the dhcp to the local setup.. this will then cause the error to be no IP instead of double NAT.. but it will restore faster and without your intervention when it returns..