Constant power discharge of a non-ideal capacitor
The equations were solved by others here. Unless I've missed a sign somewhere, this formula gives the time it takes for a cap to reach internal voltage V, starting from voltage \$V_{0}\$, with a given ESR and capacitance, and a fixed power discharge.

$$t(V) = \frac{C}{4P} (V_{0}^2-V^2 + V_{0}\sqrt{V_{0}^2-4PR_C} - V\sqrt{V^2-4PR_C}) + CR_C(\ln\big(V+\sqrt{V^2-4PR_C}\big) - \ln\big(V_{0}+\sqrt{V_{0}^2-4PR_C}\big))$$

Note that since V is the internal, unloaded voltage of the cap, "behind" the ESR, to find the time it takes for the cap to reach a specific terminal voltage while loaded, we must use the substitution: $$V=V_{min}+\frac{PR_{C}}{V_{min}}$$ where \$V_{min}\$ is the minimum desirable terminal voltage.

These calculations appear to match our numerical estimation methods nicely.