I think it was Meghan's behavior and not her skin color that is a problem to the British people. In a country as racially and ethnically diverse as Great Britain has become, it takes quite a lot of effort to earn a 98% negative response on the question of Markle's impact on the Royal Family.
In that poll taken in August of 2022 of 13022 people polled only 224 considered Markle as having a favorable impact on the Royal Family and 66 were undecided. In other words, assuming the poll was a legitimate poll, over 90% of the people of color who were polled had issues with Markle. That is pretty amazing.
Keep in mind, initially, the same public loved Harry and Meghan with a 67% positive rating upon announcement of their engagement.
So, assuming Meghan's race has not changed, it must be her actions and not her skin color that has turned the British public and royalty against her. I suspect Kate views things from how Harry's and Meghan's behavior impact the public perception of the Royal Family and now that we know the answer to that... Kate's lack of fondness for Meghan is perfectly understandable.
You may hold a different view regarding Harry and Meghan's behavior and what is expected from the Royal family. Pretty sure selling your life as a "reality" story in a country where such tabloid behavior is frowned upon compared to the US where such behavior turns people into media stars, accounts for the difference in attitudes far more than skin color.
So, neither the skin color or gender argument really fly. Let's just say, Markle's behavior regardless of her gender and race simple isn't the Brits "cup of tea".
As for being a strong woman, I seem to recall Britian has a pretty long history of strong women from Boudica, Queen Elizabeth I and II, Queen Victoria, and of course Margaret Thatcher.