7.6.1 + join wireless network + allow ethernet clients = problems

I don't have an Xbox, but if I use another computer connected to the AirPort Express which is set up to "Join a wireless network" with "Allow Ethernet clients", I cannot duplicate the error that you are seeing.

The computer always gets its network address with no problems.

What happens if you connect a computer to the AX as a test?

I still cannot duplicate this here, even after trying several devices.

I would try a complete power cycle of the network if you have not already done so as follows:

Power off the entire network....all devices....power down order is not important

Wait a minute

Start the modem first and let it run a minute by itself

Start your "main" router next the same way

Start the AX the same way

Keep starting devices one at a time until everything is powered back up

Test the network again

If still no luck, I would try a Factory Default Reset on the AX and then configure it again

Resetting an AirPort Base Station or Time Capsule FAQ

If still no luck, about all you can do is try to reload the firmware again.....sometimes it does not "stick" on the first update.....or, revert back to the previous firmware version

Replaced the Ethernet cable yet?

It has worked for me in similar situations when I never would have thought that it might be the cable. Once between the modem and the router and another in a similar remote setup like you have.