I'm trying to understand your question. I hope I got it right.
As said in the comments and here, KL is well defined between a continuous distribution $f(\theta)$ and a discrete distribution $g=\delta(\theta)$, by using the definition
$$ \text{KL}(g||f)=\int_{\Theta}\delta(\theta)\ln\frac{\delta(\theta)}{f(\theta)}d\theta=\\ \int_\Theta\delta(\theta)\ln\delta(\theta)d\theta-\int_\Theta\delta(\theta)\ln f(\theta)d\theta=\\ 0-\ln f(0) $$
the first element is clearly the (minus) entropy of the $\delta(\theta)$ while the second term is equal $\mathbb{E}_{\delta(\theta)}[\ln(f(\theta))]$.
I don't see the problem of using a conditional distribution as you did. The KL in that case, is equal to $\mathbb{E}_{\delta(\theta)}[-\ln f(\theta|y)] = -\ln f(0|y)$