In BvS we see people thinking what a threat Superman is to earth. But in Supergirl's S01E19 we see he falls under Myriad's sway. So why doesn't Non use him to do his bidding?
He could have used him to keep his enemies from stopping his plans.
tl;dr: There is no need.
Why would Non need to?
He is just as Kryptonian as Clark, what could Superman do for him that he cannot just do himself? Technically, Non could have used him to keep Kara busy, but - like he tells Indigo - he could have killed her himself (meaning he doesn't need Kal-El for that). He thinks nobody can do anything to stop him.
Sure, being forced to fight her cousin might have been traumatic, but so was what Non did to her. Arguably, what Non did instead was even worse, because it - at least for a bit - made her think it wasn't even worth the fight. She is so distraught, that when Max suggests his plan (and they discuss that Kara will have to leave the city), she doesn't even remember Clark is still in the city and would be killed along with all the other Kryptonians!
Why use Alex and not Clark?
Blood bonds us all might be a Kryptonian saying, but making Kara fight her sister and not her cousin is most likely for the reason, that Non will get satisfaction either way: Either Alex "wins" and kills Kara and Non will have free reign, or Kara "wins" and he breaks her spirit even further plus his wife's killer is dead. Kara and Kal might be biologically related, but she barely knows him, whereas she and Alex grew up together... as Max said, it's nature vs. nurture. From Non's perspective he wins either way. And nobody (least of all Non, I would imagine) knows Indigo's endgame.
Which enemies are trying to stop his plans... and why should Non care?
Non has the DEO under his control. While Kara managed to prevent the prisoner release, he has basically no enemies left within the city. There is only the US military (under General Lane's command) outside the city, but they cannot enter, without falling under Myriad's control either. They could try sending in missiles, which Non could shrug off with a bored snarl. And it is unclear if General Lane would order such a strike, considering his daughter is one of the controlled - he might, but that would not change the fact they apparently have no Kryptonite ready to fling at Non, which is why Sam and Max are working together.
Why doesn't Non go the BvS route?
The people already think exactly what he wants them to think. He has absolutely no need to make Superman the bad guy in their minds. He controls their minds. He could make them fear Superman instantly, but there is no point to it. If they spend their existence in fear, they are not productive, and they need to be: after all, they are all supposedly working on saving the planet together, programming some weird Kryptonian algorithm (or maybe that is Indigo's doing?)
Out of Universe
The (real, out of universe) reason is of course the fact they cannot actually show Superman (except from the back or the distance or on text-only IM). That is a studio decision that they can drag out almost beyond the breaking point (and forcing them to the ridiculous scenario that a rampaging Kara throwing people off buildings would not draw her cousin over from Metropolis to slam some sense into her / her into the ground, without his buddy J'onn having to out himself; hence the little exchange of Kelex informing Kara that Clark has matters off-planet...).
Mostly because the studio couldn't show him and let's face it he could have killed her if he wanted. Realistically using superman would have been ideal except knowing superman he most likely would have snapped out if it if he was forced to hurt someone then he would have had to deal with a pissed superman which if you factor in the fact that he's not only a natural birth but also he's been under the yellow sun longer would not have been good