Thanks.. that is helpful.
I do also need to see the setup on the computer.
But let me suggest some things to try.
NOTE .. when you have one computer working fine and the other failing.. the most likely cause of the issue is the computer.. not the TC directly.
NOTE .. some changes will do nothing.. but we need to track down the cause which is difficult.
Your setup is very typical.. and no big issues.. so what has gone wrong here is bound up with overlapping OS upgrades.. in the end they do funny things to a computer.
Here is the simple stuff.. Test this and report back.. I have more complex arrangement.. but lets see if simple works.
Numbers points are not working properly.. so I am going to do it with letters.
A. On the Internet tab of the TC change to static.
It is currently set to DHCP.. ie it automatically gets an IP from the ATT router..
When you select static it will basically move the IP it currently has and all the other details to static.. and they won't change thereafter.
Take note of whatever the value is at the time.. Here is the first big problem.. it keeps changing IP. Routers should be static.
B. You have a domain issue.
Look carefully at the domain on the Internet tab of the TC.. and also on your first shot of the broader network.
The domain is attlocal.net
Now look at the second screenshot.. your wirelessly connected laptop has a domain of .local
This is wrong.. and is a bug in the airport. I do not even understand how it is possible with the Airport in bridge and the ATT router handing out addresses..
Go to your computer.. I am pretty sure you will find on the wireless network preferences DNS tab you will have a search domain of attlocal.net
But your computer is in wrong domain.. Please remove attlocal.net and make sure that page is blank.
On the DNS side it should show 192.168.1.254
If it shows ATT dns servers.. replace them with 192.168.1.254
(However this is not a good idea if you are using your laptop outside home and connecting to a different network.. so consider this a test).
C. Fix IPv6 link-local.
In the earlier post I showed you how to do that on your laptop.. did you do it??
You can also now do it on your TC.. go to the Internet tab, Internet Options. Change it from Auto to Link-local only.
Test and see if you get any improvement.. this might take a couple of days.
Even if you don't immediately get an error does not mean it won't happen.
If it fails.. open a terminal and ping the TC.. just type
ping -c 20 192.168.1.87
That is just to limit the pings to 20.. that ensures the TC is awake and running.
I would then mount the TC hard disk in Finder.. you don't need to do anything except open it.
Then see if Time Machine will work.
You can backup directly to the IP address.. but this does not carry over after a reboot.. which makes it a pain.. but if you would like to, again more of a test ..
With the TC static IP.. in Finder use Go, Connect to server.
Type in AFP://192.168.1.87 (or whatever your TC static IP happens to be).
When requested put in your password for disk access.. usually just the main password. store in the keychain.
Then go to Time Machine.. click on add or remove backup disks.
Add the new disk which should show up as a different disk to the TC current name.
Do not remove the existing disk.. and put in your password.
Time Machine will now (120sec delay) check the disk.. find the existing backup and continue with the next incremental.
Tell me how that goes.
Depending on results we can go more complicated .. if it still fails.