What came to my mind – similarly – was “False News.” It even sounds close enough to the ‘original’ that you could claim that you were being misunderstood 
Might be useful to remember that CNN is an acronym – Cable News Network – while Fox is just a name, not an acronym. The all-caps usage is technically incorrect when upper and lower case text is being used, though their logo capitalizes both “Fox” and “News”. The name is a carryover from 20th Century Fox, which in turn was named after William Fox, the founder of an earlier company.
So rather than trying to invent a “backronym”, I’d just go with “Faux News” or the even simpler and more direct “Fake News”, which it certainly is. You might direct your dad to our famous “Is Fox News really all that bad?” thread in GD (answer: yes, with probably hundreds of examples) or the very many internet critiques of its lying and idiocies from reputable analysts.
Or, make it a verb - Gee, Dad, you got Foxed
Fox “News” with News in air quotes, said with as big an eye roll as you can muster.
Yeah, When I get into political fights with Fox viewers I refer to it as the propaganda arm of the Republican Party. “Fair and Balanced” is the “Pravda” of the 21st Centuty.
Perusal of the interwebs also shows “Faux Noise”. I think this might suit your needs!
What is Fox and the Kremlin? Doing more to advance mother Russia and destroy America than Trump
How about Full Of Xhite?
“MSDNC” is what he hears it called on Fox.