Buttercream is pretty simple, start with room temperature butter, beat it until it's white, slowly add icing sugar, add small amounts of milk as needed to keep the consistency right.
For each 1/4 cup (1/2 stick) of butter, use about 1 1/2 cups of icing sugar (sifted) and 3 Tbsp of milk. You can do it in a mixer, a food processor, or just use a bowl and a whisk. I used a whisk a few days ago when I realized my large stand mixer is overkill for the small amount of icing I wanted and I found that microwaving the icing for about 10 seconds helped me stir in the sugar when it was getting too thick.
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.
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I saw no degradation in image quality (custom printed ones) in a few days. If the icing is hard then you moisten it just enough for the rice paper to stick before letting it dry again you should be fine as there's no source of moisture to make the ink suffer. But of course check the instructions - I think mine used alcohol-based inks.
But why not ice the cake in advance then add the topper the day before serving?