No solid test data here, but a note or two about your assumptions.
First, the rate of heat loss is generally proportional to the difference in temperature between the object losing the heat, and the surroundings. Assuming it isn't blisteringly hot out, your talking a linear increase in cooling as the disc heats up. If both discs in your example were the same temperature throughout the thought experiment, you could just talk in terms of degrees of cooling per second, but that approximation doesn't really work here.
Second, the amount of heat the two absorb per unit temperature increase is the heat capacity, as you know. But the amount of heat they need to absorb is the same as the braking power. So if the smaller one has truly 10% less braking power, there is 10% less heat being dumped into it.
I suspect that when it matters, cooling is much slower than heating, and can almost be neglected in the calculation. Rate of cooling has more to do with how soon you can dump more energy into it. Which matters too, but is less of a factor in determining whether you're getting close to catastrophic failure.