There's quite a few reasons to disbelieve it. First, Meghan either deliberately misrepresented or hugely misunderstood the legal effect of her pre-wedding exchange of vows with Harry. Second, even if the conversation about the baby's skin tone happened exactly the way Meghan said it did, it's still easy enough to verify that it wasn't the reason for decisions about titles and security, so that's another huge misrepresentation or misunderstanding on her part. Third, it's not even clear what Meghan is claiming happened. First she said that there were multiple conversations during her pregnancy, with the clear (and probably deliberate) implication that those multiple conversations were all between her and relatives commenting about the baby's skin tone. Later Harry, seemingly horrified that it had been brought up at all, said it was one comment made to him when they first started dating, that he later passed on to Meghan. The multiple conversations had been between Meghan and Harry, all regarding this one conversation Harry had with someone else 1-2 years previously - which is yet another huge misrepresentation on Meghan's part. Knowing all of that, it would be foolish to take anything she said at face value, especially regarding a conversation she only heard about secondhand.
Notwithstanding any questions about Meghan's honesty and accuracy, I think that context makes it even less likely that it was racist or at all offensive for one other reason. According to Meghan, Harry knew she was suicidal at that time. Are we supposed to believe that he thought it would be a good idea to tell his suicidal pregnant wife "Just FYI, my great aunt Duchess So-And-So told me two years ago that she'd never speak to me again if I had a brown baby"? Even if that conversation with Duchess So-And-So had happened, passing it on to Meghan at that point would only have needlessly upset her. Why on earth would he have done that? The only explanation that makes sense here is that Harry perceived the comment as innocent, and expected Meghan to see it the same way, and then she didn't. That doesn't mean she was wrong to feel differently about it, but it couldn't have been all that bad.