Mac Mini private network
I didn't see you mention iTunes at all in your post.  iTunes is essential.

Unfortunately, AFAIK, you can no longer just "share" movies from iTunes to ATV.  You have to turn on Home Sharing, and I believe this requires a live internet connection.  Both iTunes and ATV log into Home Sharing, and everything loaded in iTunes is available on the ATV.

If you do not have an internet connection in this setup (and can't/won't add one), you can setup Airplay Mirroring on the Mac mini (2011 or newer, I believe), play the movie in iTunes, and send the content to the ATV.  This is not ideal to me, as you have to use the Mac (iTunes) to control the ATV.

Home Sharing is the way to go with ATV.

There are lots of other options like Plex if you want to use Roku or Chromecast, which should work without an internet connection.