Capacitor charging efficiency with a constant power source
You are asking about efficiency and I am not able to answer that, but maybe it helps to answer the question of the capacitor voltage U(t) for constant power charging without considering R_ESR. 

With constant power P, energy E over time in the cap is 
$$ E(t)=P*t=\frac{1}{2}CU(t)^{2} $$

This can be rewritten as
$$ U(t)=\sqrt{\frac{2Pt}{C}} $$

We can also find I(t) with following equation...
$$ I(t)=\frac{d}{dt}CU(t)=\frac{C}{2}\sqrt{\frac{2P}{C}}*\frac{1}{\sqrt{t}} $$
And here is the graph for U(t) [green] and I(t) [red] with constant power P=1W and C=1F. 

This is the voltage and current of the capacitor when it is charged with constant power source.