Use VLC on the Mini.
In a situation like this, it's far simpler to 'pull' from the older Mac than try to 'push' from the newer.
All you need do is set your MacBook to share the media drive. System Preferences > Sharing. Share your Media folder to an account you can allow to connect.
Once shared, then from Finder on the Mini, go to the Go Menu, Network & choose the shared media drive. You may have to start from the top level on the computer & 'drill down' first time, but it should appear in the list as a shared drive once connection has been previously established.
Find your file on the remote Mac & just drag directly into VLC's playlist window. It won't copy it, it will just play it from where it is - so you have to make sure your Macbook doesn't go to sleep while it's playing. VLC will keep the Mac it's playing on awake, but it can't also keep the remote server awake.
Tap spacebar [Play].
Movie on demand, no 'casting' required.