ATT Uverse Airport express

I can't turn off the Wi-Fi network in the ATT U-verse gateway.

Not sure what this means. I have yet to see an ATT gateway which will not allow the WiFi service to be turned off on the device. Are you saying that you cannot turn off the WiFi on the ATT gateway because it does not have a setting to do this, or are you saying that that you don't know how to do this? Or, are you saying something else?

Currently the Airport express is set up in bridge mode on the 5g band of the ATT Uverse Wifi network.

Problem: The wireless clients are swapping back and forth between good signal in living room and fair signal in laundry room.

This is a common problem on networks with multiple WiFi access points, especially when the access points are from different manufacturers.........and one of the reasons why newer mesh systems are selling so well. The mesh systems in general are a lot better about keeping track of which clients are connected to which WiFi access points and changing things on the fly.

Would changing the radio channels help this situation?

No.

How do I get the wireless clients to tell the two base stations apart on the same network and stick to the better signal?

Unfortunately, there are no settings on WiFi clients that would tell the client which WiFi access point to connect to at any given time.

I had the Airport express set up as a secondary network, which worked except for the double NAT error

Then, you might think about ignoring the Double NAT if everything else worked the way that you wanted, except for the Double NAT. AirPort Utility provides the option for users to ignore the Double NAT and the AirPort Express will display a green light.

Personally I would like to see him use a switch on the LAN side and a USB to ethernet adapter for the amazon fireTV stick to simplify things dramatically

Does this mean that the AirPort Express would not be used at all in this type of setup? If yes, you do have the option to turn off the WiFi on the AirPort Express and it will function as a simple Ethernet switch when it is set up in Bridge Mode.