Are you sure you want to use icing? The barbie cakes I have seen all have the skirt made from rolled marzipan, not from icing. The bodice can be a normal fabric top or dress (inedible, can be washed after the cake is eaten) or also molded from marzipan. I agree that it isn't as tasty as buttercream icing, but it surely makes a more beautiful skirt.
This recipe has 196 pictures, you can see different variations of the bodice.
If you insist on using icing, I would try to smear something sticky onto the doll first, for example honey or syrup in the softball stage, then apply the icing on the sticky layer. I haven't tried this myself (nor have I tried applying icing directly to plastic, so I don't know if it will hold).
On another note, don't forget to put the barbie's legs in a cut-off paper towel cylinder, so she doesn't get damaged when the cake is cut, and doesn't fall when pieces of the skirt are missing. Also, if you bake the layers in a guggelhupf pan, you don't have to cut a piece out of the middle, which is hard to get right.
Perhaps you're looking for the soufflé cheesecake, which has a moderate amount of flour in it. This style is also popular in Japan.
The other style popular in Japan is the "rare cheesecake", which is set with gelatin instead of being baked. I think this is probably denser than what you're referring to, so I left it out of my first edit, but worth considering if the soufflé style doesn't match your expectations.
In Germany, some quark cheesecakes have flour in them as well, and those often have a more sturdy texture with a bit more air than the New York style.
Best Answer
If you want a sculpted cake of this type:
then I would advise against trying it at home. This is a skill you have to learn over years. Basically, you have to be an experienced sculptor in working with cake materials, which are harder to use than the typical art materials like clay. Look up "cake sculpting" on YouTube to see what they are doing. Or search for Cake boss episodes, taking into account that this "reality" TV is heavily edited - they are showing you a very clean and quickened version of the process. Really, I think that 500 dollars is cheap for having your drawing sculpted. By the way, the airbrushing tool alone used for finishing the cake in the picture runs at around 200 dollars, although of course you don't strictly need it for making the cake in such a shape.
You seem to be good at drawing. If you can make a finished, colored drawing out of it instead of just a sketch, you can go to a bakery who has a foodgrade printer, and have it printed on the cake. They mainly advertise it as "your cake with your photo on it", but of course they can print a digitized drawing too.
If you don't want a picture, it is also doable as a 2d picture piped onto a flat cake. If the 500 dollars are for a sculpted cake, then having the picture piped can be much cheaper (I don't think that they would have required 500 dollars just for piping, but who knows, maybe you live in Manhattan). It would be this type of cake, but with your drawing on it:
If you have steady hands and some artistic experience, this kind of decoration is a skill that can be learned in maybe 20 hours. Take a piping bag, make icing, and practice on other surfaces first, even stretched plastic wrap. When you are good enough, attempt the drawing on a practice surface. Then you are ready to make the cake. Fondant or marzipan will look best as the background for the piping, but they are very hard to work with. If you have never done it before, use a cake frosted with buttercream as the background. If you have the time and inclination, you can learn how to make marzipan flowers and add them as 3-d elements on top of your drawing.
The basic skills you need are:
The Internet is full of tutorials on decorating cakes, watch YouTube videos about the more visual parts. There are good articles on making icing too, and we also have one or two older questions on that.