I dont think the PR department would have much of an issue at all. Its not like they are superstars, since when do you know of let alone follow the lives and families of CEOs or billionaire owners unless they are the kind that seek notoriety? Just as you said, they could make someone else the face of the company or just one of step sibling pair. There is no requirement to enter high society and publicity.
Its only now the pair as sudden heirs are interesting to the general media, the sniffing of blood such as an inheritance quarrel and thus potential for implosion of the company. If there is no destruction the media have nothing to gain. Alex as the mystery wayward off the radar son only matters in this moment because he is an unknown until a new status quo is reached. No quarrels among the heirs and soon everything settles and they will only matter to economic and medical media.
As you said the only potential PR issue they face are getting "caught" early on and then acting like its something shameful and wrong. But that can be fixed/mitigated with some prep and smooth continual operation of the company. Get privately married and live happily ever after.
As long as they don't lose the case, company closes or something else of that nature, I don't think they have to worry about media backlash all that much. Besides, I don't think they will lose anyway, seeing as how Prizer make controversial moves like a Bond villain.
Like it has been already said - what MC and Jaye are doing is neither forbidden, nor is it illegal. Sure, there will be probably some media scrutiny, but it won't be something they won't overcome. They don't have to lead the company themselves anyway - Lisa seems to be doing an amazing job and getting rid of her is going to be a mistake anyway. Besides, the owners of companies of that size aren't involved in the day-to-day operations of them anyway. Like the owner of the company that I worked for in the past, I've seen like a handful of times in the 5 years I worked there. He would show up in a helicopter for the yearly quarter meetings of the Board of Directors to see how the company is doing and then he'd take the helicopter back to the airport, hop on a plane and go live his ilfe.
I'm not sure there are actually laws on the books (anywhere) that would specifically cover the case of clones and incest. Sooooo....
It will certainly be interesting how the devs handle that part of the story. But I believe in them, as both of them are talented enough to come up with something that fits the story.