its not dns broadcasting... It would be the client doing a netbios broadcast for the hostname... Hey who is called somehost..
So your clearly not doing dns redirection.
So you want to set it up correctly.. Point your clients to pfsense, or some other local NS that will resolve all your local resources, and will then forward or resolve all your public dns needs.
Pointing clients to outside NS is not going to allow you to actually resolve any local resources, nor will it give you the ability to block bad stuff.. You have no control over the dns at all when you tell client to use 8.8.8.8 for their dns..
But if you have them point to something local for dns, say pfsense - you then can control stuff by blocking stuff you don't want them to get to.. You can resolve say www.whatever.com to the local IP its hosted off of, vs getting the public IP for this fqdn and having to use nat reflection.
Also pointing clients locally allow you save some bandwidth, because if client A looks up www.something.com, and then client B asks for it its already cached at your local dns, and doesn't have to be looked up again, etc.
So fix it already - not really sure why we are stilling having this discussion ;)