In a bowl, combine the orange juice, Campari, rum and shredded beets; refrigerate overnight. Strain the beet mixture through a fine sieve into a small saucepan, pressing on the solids. Whisk in the potato starch. Simmer over moderate heat, whisking until thickened, about 1 minute. Season with salt, pour into a bowl and let cool.

In a saucepan of boiling salted water, cook the daikon until crisp-tender, 8 minutes. Drain and let cool; pat dry. Halve each piece of daikon lengthwise, then arrange the pieces on a work surface, cut side down. Make very thin lengthwise slices, stopping 1/3 inch from the top. Stand the daikon fan, sliced ends down, in the beet infusion until the ends are deep magenta, about 1 hour; occasionally spoon liquid over the top.

In the small saucepan, melt the butter. Add the red onions and a pinch of salt. Cover and cook over low heat, stirring, until very soft, 15 minutes. Scrape into a blender and puree. Return the puree to the saucepan. Cook over moderate heat, stirring, until thick, 10 minutes; season with salt.

Carefully drag a daikon fan across each of 6 plates; set on the plate. Stir 1/4 cup of the beet infusion into the warm basmati rice. Mound the rice and the onion puree on the plates and serve.