badtlc4 IPv6 is a different beast. And managing residential IPv6 can get complicated.
There are basically two ways hosts can acquire an address with IPv6. There is a “managed” method using DHCPv6 that works similarly to DHCP with IPv4. There is also a less managed method called SLAAC, which is where hosts can self-assign addresses using information that is multicasted on the network in a router advertisement. You can even run both concurrently with the correct configuration. The distribution of network information is also typically done with the RA regardless of whether you use SLAAC or not.
I’m not sure how the BGW’s IPv6 configuration is set up or how configurable it is. Or if the BGW allows you to configure the DNS servers advertised in its own RA service. But it might be trickier to use your own RA and/or DHCPv6 services with the dynamic addressing situation with IPv6 regardless.
In short, the ideal situation is where you can configure the BGW’s RA service to multicast the DNS servers you want.
If that’s not possible and you’re comfortable managing the IPv6 configuration on your own, including when the delegated prefix that you get from AT&T changes, you’d need to disable (or drop using a firewall of your own behind the BGW) the RAs and any DHCPv6 from the BGW. Then you would need to configure a router advertisement service with the network prefix (should be a /64 address block per network) that AT&T is giving you, the default gateway (which will be the link-local address of the BGW LAN interface), and the desired DNS servers. You can run that alone to do SLAAC. If you also want to do DHCPv6, then you need a separate DHCPv6 service that will also hand out addresses on your assigned prefix.
I’m a big advocate for IPv6 adoption. But in circumstances like these, my general advice to those who aren’t comfortable managing a full dual stack (IPv4 and IPv6) environment, is that it’s generally best to disable IPv6. IPv6 introduced a lot of complexity that makes it more flexible and adaptable to more needs. But with that, unfortunately, simplicity was lost.