Does Bell's theorem imply a causal connection between the measurement outcomes?

It kind of does, but in a useless way.

The question is essentially equivalent to the following simplified version of it: suppose a probability distribution $p(a,b)$ cannot be factorized as $p(a)p(b)$. Does this imply that $A$ "causes" $B$?

The answer is: not really. The problem is that, from a purely probabilistic point of view, there are no "causal relations", only correlations. Whenever there are correlations between variables, their marginals can be written so that one variable "looks caused" by the other, but this statement does not have much value.

Indeed, we can always write $p(a)=\sum_b p(a|b)p(b)$, which makes $A$ being "caused" by $B$, because $p(a|b)$ is defined as $p(a|b)\equiv p(a,b)/p(b)$, and the marginal $p(a)$ by $p(a)\equiv\sum_b p(a,b)$. While true, this is not a very useful observation.

It is meaningful to talk about causation in a context in which one can actually exploit such correlation. For example, if one can choose to have the variable $B$ assume the value $b$, then it is meaningful to talk about the conditional probabilities $p(a|b)$. Notably, this is exactly the kind of thing that we cannot do in quantum mechanics: the measurement results are probabilistic, and therefore cannot be controlled.