Weak Bayesian evidence is neither necessary nor sufficient for a fallacy in this framework.

There are arguments which provide strong evidence that are still fallacious. As an example. 1% of the population considered is B. 90% of A are B. Therefore you should be 99% certain that X is a B, because you know that X is an A.

There are arguments which provide weak evidence that are not fallacious. As an example. 1% of the population considered is B. 25% of A are B. If you learn that X is an A, you should adjust your probability that X is a B upward.

The key to many fallacies is not weak evidence. The key to fallacies is evidence being treated as stronger than it is. This has the interesting implication that most arguments that claim complete certainty are fallacious.