So it does pull its IP from the RG via DHCP in that passthrough mode, just the public IP? In that case it should act the same as the modem, with a very short lease time and the RG should hand it its new IP when it gets one. That apparently doesn't work right on the RG (damn things are buggy as hell if you try to do anything other than using it as your NAT device, so that wouldn't surprise me in the least). The firewall you put behind it isn't "detecting the IP change", the RG has to assign it the new IP via DHCP and such devices generally do so quickly by assigning very short lease times. If it's not handing out very short leases, it'll take time until the lease is renewed and the new IP picked up. Doing double NAT isn't the best thing in the world, but I would expect that to behave better on the RG, and its regular DMZ mode seems to work fine. There isn't a functional difference between the two.
If I've learned anything in having the misfortune of working with those Uverse RGs on mine and several customers, it's do what works on the RG and be glad it's working. From its crappy stateful firewall that can't be disabled even with a static IP assignment (disable firewall doesn't disable anything), to numerous bugs throughout other things, those RGs suck.