I'll start with my take, bearing in mind I am not a Japanese speaker, and you may wish to consult with a Japanese Speaker for your legal and comprehension needs.
I mean where do you start? The first thing to note is that when you pick up Shika for your secret production runs, is that she'll ask for 30 minutes first, so she can slip away first.
Bearing in mind you're taking her to a concert for her to perform in front of thousands of people.
So this leaves a series of questions:
a) She might be busy, and even if the producer can plan things meticulously to have her ready to go to perform when she gets there (not happening cause she needs to get stage makeup and prep and that takes time) and that wastes all the resources the producer has to outlay for these events to happen. (Fortunately, the president is allowing the use of resources for this, or otherwise our P would be an embezzing P, and will be spending time practicing the jailhouse rock if court)
b) Kuroi IS going to notice his own daughter suddenly getting popular and writing in mail and other oddities. Nevermind the producer being the best business ninja ever - you don't HIDE that sort of thing forever. I mean the producer's job during this arc is to help her make the extreme live (which means you do Kuroi's job for him, more or less) and... Kuroi doesn't really notice?
c) Could you imagine the conversation? "Dad, I need to do something for the next few hours, mind if I go?"
"You're scheduled for vocal practice in an hour, you're not going anywhere."
"But please?"
... Seriously? I somehow doubt that Shika making puppy eyes to her dad's going to convince him to let her disappear for hours on end.
Then there's the legal problems.
If I remember right, Shika isn't that old, so there's the big problem of age - As her legal guardian, he could charge the producer with kidnapping. He could allege poaching and contract breach if she's formally part of the 861 agency. Doing this LITERALLY gives Kuroi the ammunition to destroy 765 with so many clauses that it is business suicide to go ahead with this.
If nothing else, by law, he could instruct 765 from ever getting near her for business purposes, and legally 765 has to obey the moment they become aware of her identity as long as the request was legal and reasonable.
The president is AWARE of the arrangement - he's there when she asks the producer, and agrees and lets the producer be a secret producer (He suggests the title in fact), and allows him to do work for her while letting 765 run itself for a while.
Uh, mister president. Corporate responsibility for your COMPANY?
Then there's the relationship issues - nevermind the grudge and the rest of it, Kuroi's had enough intimacy to have a KID. If the President KNOWS about it (and it's suggested he's aware) he's either a moron (see previous point) or he's doing it to score a cruel point against Kuroi (which works tactically, if you ignored the previous point) in their feud.
Of course, I don't know much more - her mother isn't discussed. Cue the fanfiction I suppose?
In short: At first, second and third brush, the sequence of events that occur given the facts given at the end of her arc is... a bit fantastical, considering im@s is a heavily romanticised (Namely, it's buffed up and shiny and the real industry is much more cutthroat) but actually fairly realistic simulation about how the idol industry works. There's absolutely no way in HELL you could have the above happen without spending time doing the jailhouse rock.
