The way I understand it the group should have filed the complaint with the cable companies (Spectrum, Comcast, etc). Since it is a subscription service the FCC doesn't have authority in these instances (it's not going out on public airwaves).
But then things get a bit sticky. If an actual lawsuit grows out of this you could get into things like 1) people rent the box that receives the signal 2) news is defined as "serving the public good" 3) intentional deception. Oddball notions such as Net Neutrality could even enter the picture.
Although I'd like to think that CNN could be forced to drop "News" from their network name I can't imagine it ever happening. Still, forcing the issue raises public awareness and that's always a good thing.