For lots of things double NAT is fine.. and when I am testing routers I am constantly using double NAT.. but on anything but testing it is to be avoided.
1. The primary issue is the IP address is not stable on the Time Capsule (I will use TC=ATC)
So set this statically. I recommend you do it on your main router via dhcp reservation but you can also very simply do it by changing over from DHCP to Static on the Internet tab of the TC.. Or even do it on both..
2. Domain is important. If your computer is getting a strange search domain via the main router, then it will be unable to find the TC when it wants to backup. Sometimes you can reset this in DHCP for LAN on your AT&T router.
Sometimes you will have to fix it manually on the computer.
3. Apple depends on IPv6 being set correctly. Unless you are running your whole network on IPv6 the easiest solution is set both the TC and the computer to link-local only. Auto is the wrong setting here.
4. Do not use APPLE RECOMMENDED NAMES. Since Apple moved to SMB as the main protocol for network way way back but much more seriously in Sierra and High Sierra. They never fixed the naming recommendation from AFP which was the older Apple default protocol.
So keep names for everything short, no spaces and pure alphanumerics less than 16 characters.. bonus brownie points for <10. Passwords use the same rules but can be longer.
5. With your MacBook Air issue.. you can mount the TC in Finder first by IP instead of by name.
And since Apple MUST use AFP protocol to the TC I think you will have to use AFP.
Go, Connect to Server.
AFP://TCIPAddress (where you replace TCIPAddress with the actual IP now static on the TC)
But there is another method entirely. More coming.