Success in destructive testing-- or, I broke it.
Looking into annealing the sugar models by putting them in a warm oven, near the melting point. These were put in a hot oven, close to but below the melting point of sucrose. (A digital oven thermometer is a must.)
After half an hour in the oven, this was the result: a puddle. The bottom of the models had melted, and the lower "legs" of the model had softened significantly as well. On the top side, there was indeed some evidence of smooth annealing. I suspect that some of the melting is due to the cookie sheet- perhaps it was warmer than the air in the oven-- the same thing happens to chocolate chip cookies; they can get easily burnt on the bottom. The next thing to try might be using a semi-insulated cookie sheet.