Do you ever feel like you're tip-toeing around her, altering your behaviour in order to appease her? If so, stop. You say that you've only gotten more time than her a few times in years... So be more assertive, and get more time with him. Then let her keep score, let her complain that you spend more time with D than she does. And then say to her: "So what? Deal with it. You're not
#1, and you are not entitled to special treatment. Get over it, or find someone who wants a #1." The only reason she continues these outbursts is because they're working for her. The only way to make them stop is to quit allowing them to work for her. Even that doesn't guarantee they'll stop, but you can be sure they won't stop as long as they're creating the outcome she desires.
People will tend to behave the way they always have, until they begin to behave differently. If Ka shows no signs that she's going to stop asking, quit hoping that repeatedly requesting it will magically make it happen.
I agree with previous posts that it's D who's dropping the ball here. After a certain point, the "he's new to this" argument stops holding water. You've told him explicitly what you want him to do: Stop sharing details about your relationship with Ka. There's nothing about being new to poly that makes that unclear or especially difficult. There is something about being a passive person that makes that difficult, and if that's how he is, then it's not likely to change. Don't make excuses for him.
Unfortunately, from what you've said, it doesn't sound like the situation is likely to change. You're doing everything you can to cope, and it's not enough to make you feel comfortable. You might need to consider that Ka's baggage is "the price of admission
" for being with D, and then decide whether your relationship with D is worth that price.