Caroline doesn’t have a claim to any of Matthew’s money. I have to go back and listen to the scene where they read his letter, but I was under the impression that “you must take charge” meant she was his heiress for the purposes of being in charge of George’s half of the estate (i.e. Matthew’s half which he left to his son). In other words she’s just the trustee with a fiduciary duty to George not to do anything crazy with his money until he’s of age (like leave it to a stepsister). I mean I don’t think Julian Fellowes knows what he’s talking about because the letter is saying two different things: “you’re my sole heiress” + “take charge (of the estate) for our child’s sake”—like which is it? The debate wasn’t who was ultimately going to get the money because the answer is George, just who was going to look after his share until he was of age. None of that money specified in the letter acting as a will would Caroline have a claim on. Or at least that’s what I would argue if I was George’s lawyer at any rate.
She’d probably get something but that something would come from Mary’s allowance or Henry himself, not from Matthew Crawley’s estate.
EDIT: So I went back and read a transcript and she is the beneficiary, and I’m assuming it’s fee simple so she could theoretically convey it to Caroline, but I don’t really think she would because the intent of this letter was clearly for the interest to go to their kid eventually.
If anyone’s interested: