Oh, and one addition to the direction:
While this works perfectly fine, through additional testing/experimentation within AdGuard Home, I found that you can also leave the persistent client record in place and just comment out the upstream DNS servers within the persistent client record (via placing “#” in front of the upstream DNS servers for the VPN provider) when you’re no longer using the VPN connection. It will not look like you’ve commented it out, but it will still work like commented-out content. This means the persistent client record will, in effect, show nothing for custom upstream DNS servers, in which case AdGuard Home will use the “standard” / “default” upstream DNS servers you’ve configured globally. Alternatively and for a similar reason, you can also just leave the persistent client record in place and delete the custom upstream DNS servers, and once again, AdGuard Home will just default to using the globally-configured upstream DNS servers.