Place chocolate in medium metal bowl. Bring cream to simmer in small saucepan. Pour cream over chocolate; whisk until chocolate is melted and smooth. Whisk in rum. Cover and chill at least 6 hours. (Can be prepared 2 days ahead. Keep chilled.)
For coconut meringue layer
Position rack in center of oven; preheat to 325°F. Line 17x11-inch rimmed baking sheet with parchment paper. Mix coconut, 1/3 cup sugar, flour, and salt in medium bowl. Using electric mixer, beat egg whites and cream of tartar in large bowl until soft peaks form. Gradually add remaining 1/3 cup sugar; beat until stiff but not dry. Fold coconut mixture into meringue just until incorporated. Spread meringue evenly over parchment on prepared sheet, covering completely. Bake until light golden and just springy to touch, about 20 minutes (meringue will be soft). Cool completely on baking sheet. (Can be made 1 day ahead. Cover tightly with foil; store at room temperature.)
Place chocolate in medium metal bowl. Bring cream to simmer in small saucepan. Pour cream over chocolate; whisk until chocolate is melted and smooth. Whisk in rum. Let glaze stand at room temperature until thickened and spreadable, about 1 hour.
Place sheet of waxed paper on work surface. Cut around edges of meringue to loosen. Invert meringue onto waxed paper. Peel off parchment. Cut meringue crosswise in half, then cut lengthwise in thirds, forming six 8 1/2x3 2/3-inch rectangles. Cut piece of cardboard into 8 1/2x3 2/3-inch rectangle; cover cardboard with foil. Place 1 meringue rectangle on cardboard.
Using electric mixer, beat ganache just until lighter-colored and firm enough to spread. Using offset spatula, spread 3 generous tablespoons ganache evenly over meringue. Top with second meringue rectangle; press to adhere. Spread 3 tablespoons ganache over. Repeat procedure with 3 more meringue rectangles and remaining ganache. Top with remaining meringue rectangle. Chill cake until ganache is firm, about 1 hour.
Place cake on rack set over rimmed baking sheet. Spread top and sides of cake with some of glaze. Chill 30 minutes.
Place bowl with remaining glaze in skillet of barely simmering water and rewarm just until instant-read thermometer inserted into glaze registers 90°F. Pour glaze over top of cake, allowing glaze to drip down sides, spreading evenly over sides. Using spatula, scoop up excess glaze on sheet; spread over sides of cake to cover. Press toasted coconut onto glaze. Transfer cake to platter. Chill until glaze is set, at least 1 hour. (Can be made 1 day ahead. Cover and keep chilled. Let cake stand at room temperature 30 minutes before serving.)
Using serrated knife, cut cake crosswise into 12 slices. Transfer to plates.