Why would the king let it come to light? I think it's more that he knows the child isn't his, but doesn't care and claims the child is his anyway. The whole thing about the coup isn't about forgiving Elis and more that Regis is a big believer in faith and prophecy and thinks this type of thing was destined to happen. I'm pretty sure Alania, Elis, and Lia are the only ones who know that Regis is sterile.
What do you mean why he would let it come to light? That's not the main point. The point is that he's just not going to be okay with his son fucking his wife. Fate and prophecy doesn't have anything to do with Elis fucking his mom. Even Sera herself pointed out that he doesn't believe in every prophecy that was whispered to him. And I highly doubt the King got word of some big prophecy involving incest with his son and wife.
And even on the unlikely chance that the King would somehow allow it to happen and doesn't reveal anything himself, it doesn't mean there's no way for others to find out. Hell, Valerian nearly, if not flat out caught Elis eating Elin out at the tavern. Elis isn't exactly subtle and neither is Sera. So even if the King wouldn't care for whatever reason. He's not just going to say sure, keep fucking your mom, get her pregnant and don't get caught! I'm not going to react, it's fate after all. What kind of reaction is that? Especially when the chance of someone finding out isn't exactly low.
As for the second bit, we don't actually know how many people know about "Regis" being sterile. There's many secrets that are going around in the kingdom. And it's not like Alaina killed every single person who knows of that secret. Not to mention, it doesn't take much to put together that the King hasn't been with Sera alone much at all. Especially since he was recovering. The King can try to hold his tongue, but that doesn't mean those around him won't know anything.
But hey, anything's possible, I suppose. I definitely don't think he's just going to be okay with it though.