Long for a comment but consider this to be an extended comment and not an exact answer:
At least for Poisson I can say that the random variable should fit the three Poisson postulates. Poisson RV generally describe discrete events in continuous intervals. A fitness function doesn't seem to be such a type of RV; it is a property of a population rather than an event.
I would also not expect it to be exponentially distributed which would mean that probability is highest for the most fit (or most unfit) individual i.e the more fit are more frequent. I assume that it cannot be decided who is the most fit until selection happens. This may still be the case just after a selection event but not otherwise.
It seems that this function should be Gaussian: assuming that there is little variability between individuals and what exists is because of independent events of mutation, these should converge to Gaussian as per the central limit theorem.
But generally the term that best describes this RV is a performance function. Fitness would always be a conditional RV.
For e.g. I would very roughly define fitness like this:
$f(X) = P(m(X)\ |\ p(X),S)$
where $f(x)$ is the fitness of $X$, $m(X)$ is the survival probability of $X$, $p(X)$ is performance in different tasks and $S$ is selection event.