I admit I'm not an expert on Korean noble traditions, but elsewhere traditionally, marriage was the method of determining inheritance. An illegitimate child was illegitimate because they were not recognized by law and thus wouldn't inherit. Outside of bribery, it's hard to imagine any medieval judge that would consider giving an inheritance to a commoner over a legitimate noble descendant. Even if they were called out by name in a will, the wealth would probably be put in the hands of a guarantor.
That being said, I'm still on board with this cause the characters are fun. We'll see where this crazy anachronistic setting goes.