Best way to set up airport time capsule with ATT Wireless network

What is the make & model of the ATT-provided router? Do you have the TC configured as a bridge?

By default, your TC is performing as a router. That would mean, in your current configuration, you would have two routers in series performing this function and may be the reason for the bandwidth performance issues you are experiencing.

Unless you need to keep both networks separate, I would suggest that you consider configuring both routers for a roaming type network. With this type of network each router will broadcast their own wireless network, but both would use the same network name (aka SSID) and use the same wireless security type & password. That way, any wireless network client would just "see" one network.

This type of network would also require at least two other things:

1. Each router used would need to be interconnected by Ethernet. Which yours already are, and
2. Except for the "main" router (in this case, the Arris), the other routers must be reconfigured as a bridge. This disables the router's routing capability so that just the Arris will perform this function.

Only the TC would need to be reconfigured as a bridge. The link I provided should help you understand the desired configuration for each of the routers used in a roaming network. There should be no need to make any changes to the Arris.