The argument in JKO (short for Jordan-Kinderlehrer-Otto-98) is based on a Lyapunov function for the Fokker-Planck equation.  As such, it requires that this Lyapunov function evaluated at $\rho^0$ be finite.  (The main result of JKO assumes this condition.)  This is a strong constraint on $\rho^0$: it seems to require, e.g., that the support of $\rho^0$ be $\mathbb{R}^n$.  For your convenience here is the Lyapunov function from JKO: 
$$
F(\rho) = \int_{\mathbb{R}^n} \rho \; ( V - \underbrace{(- \log \rho )}_{\text{free energy of $\rho$}} ) dx
$$

*To answer your question* and as far as I can tell: it does not seem possible to prove using standard techniques existence of solutions to the Fokker-Planck equation with what you call "nasty initial data."  Recall, in the standard (probabilistic) approach, the Fokker-Planck equation is "well-posed" if:

 - $U$ is of class $C^2$;
 - the transition probability of $X(t)$ admits a probability density function that is twice differentiable for $t>0$; and,
 - the probability law of $X(0)$ has a continuous probability density function on $\mathbb{R}^n$.