Consider a random variable $x$ and $\theta$. Suppose $f(\theta)$ and $f(\theta|x)$ are given. Suppose, in addition, that $f(\theta|x)$ satisfies the strict monotone likelihood ratio property. That is, for every $\theta ＞\theta'$ and $x > x'$, we have $f(\theta'|x')f(\theta|x)>f(\theta|x')f(\theta'|x).$ Will they uniquely determine the joint distribution $f(x, \theta)$?
No. For example, if $f(\theta|x)=f(\theta)$ for every $x$, the distribution of $X$ can be any distribution.
Edit to answer the revised version of the question There is no reason to believe the SMLR property + the distribution of $\Theta$ + the conditional distribution of $\Theta$ conditional on $X$ determine the joint distribution of $(\Theta,X)$ or, what is equivalent in your context, the distribution of $X$. If you have reasons to believe they do, you could explain why.