I have a mother that has similar antics. Actually, she is more of a martyr and a narcissist. If things aren't revolving around her then she plays the childish whiny attention seeker role really well. I totally have a hard time with that in others also. Strangely enough I picked a career where I deal with that almost daily, lol. Funny how people do that shit huh?!
I also have a hard time with competition and avoid it where ever I can. I find that ignoring it and doing whatever I feel is right for me is the best tactic. Sometimes people think that I am being competitive back but when they look more closely they can see that I have no other motive than to achieve what I want out of life while considering others as well. I do my best to not stepping on their toes and give them the respect that they deserve all while expecting the same back again. It doesn't always work, but I can't make them do stuff... they have to decide to. I just give them an example by doing what I would like in return.
One of the best experiences I had with this is that someone asked me if I was trying to compete. They came right out and asked and I appreciated that because it meant that we got to talk about it and they came to realize that I was not a threat to them, but I was not going to back away or make things happen so that they could continue to maintain their attitude of attention seeking and competing. I didn't actually say it like that but let them know what my agenda was and that I was doing things while keeping them in mind as I knew they also needed the space to do what they wanted and to be who they are. I asked that they give me the same respect and they have. Maybe this will work for you?
I wouldn't back away if I were you. I think that is a lovely thought but will be counter productive as she won't get why you are. I think I would let her know that you intend to work on making the situation be one that is mutually beneficial to both of you and that you hope she does the same thing. I think I would laugh it off as a kind of joke that the two of you need to compete for his attention because its silly, he gives you both lots and works hard at balancing his plates where you are concerned. Laugh and say, "isn't he awesome that he is so good at dividing his time and energy? I am so glad that you are so good at not causing him to work harder by asking for more attention. We are both good at not doing that aren't we....? HAHAHAHAHahahah......" *grin
Sometimes telling someone that they are a certain way, even when they aren't makes them think that maybe they could work harder on that....