if the router doesn’t allow DNS changes you are lost.. the only solution would be another DHCP server.. what I mantioned - use you Adguard as “provider” DNS of your router if possible. This way the router resolves everything from Adguard.. Look at this guide section “Pi-Hole as upstream of your router”
I think you are using your router public IP as AAAA record, which is not right - it must be the effective public IPv6 address of the server. This brings some challenges with dynamic IPs but it’s another story. maybe this helps NC not reachable over Domain (IPv6) - #2 by wwe