After searching for butterless white chocolate icing, I found an icing using shortening instead (which I am no expert, but you would probably STILL have trouble with it in the heat). Have you thought of the option of maybe using fondant icing? I found a site with SEVERAL homemade fondant recipes that don't look TOO extremely hard to make. I do know that MOST people do NOT like to work with fondant because it isn't always as "forgiving" and manageable as regular icing (my mother-in-law has been a professional cake decorator for close to 30 years). Anyway, it was just a thought. Let me know if you want the recipes for the fondant.
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Editor's note:
All About Fondant and Basic Fondant with Nine Flavor Variations from the Joy of Cooking