I’d be tempted to try redirecting anything that’s going out to port 53. You need to do this for both TCP and UDP. That way it doesn’t matter what your clients have set as their DNS server addresses.
Yes, that would certainly be an alternative. The only reason I thought the above might be better is if the DNS cache fails or is ‘off-line’ for some reason, everybody would fail over to my upstream DNS servers, if they had the correct addresses. If a client had the incorrect addresses (e.g. I fat fingered them during installtion) and the DNS cache failed, they would loose DNS service, and that would be the first time the error was apparent.
I guess it’s half a dozen of one, six of the other, unless one method is faster?