A sufficient condition for $\{P_1,\dots, P_{M_n}\}\subset \operatorname{Hull}\{P_{Y^n|X^n=x^n}| x^n\in \mathcal{X}^n\}$ to be a working codebook is that it has some $c>0$ where taking $Y_m\sim P_m$ for each $m$, then for any $\ell\!\neq\! m$ we have: $$\mathbb{P}(P_m(Y_m) \leq c\cdot P_{\ell}(Y_m))=\mathcal{o}(1/M_n).$$
Then the following will be a working decoder: "Decide $m$ was sent if $\mathbb{P}(P_m(Y_m)>c\cdot P_{\ell}(Y_m))$ for each $\ell\neq m$." For $c=1$ this is just a maximum likelihood decoder.
This is a kind of unsatisfying answer since I am suspicious that you can come up with a channel that has a codebook that can be reliably decoded, but where the decoder has to do something different than this.