If I understand it correctly, you are trying to use weights the same way
brms does (as discussed e.g. here: https://discourse.mc-stan.org/t/weights-in-brm/4278/5). In this use case, weighting just means you treat an observation with weight 2 acts exactly as having the same observation twice in the dataset.
In this case, it doesn't make much sense to use the weights in posterior predictions - if you added the observation multiple times into your dataset, it wouldn't change the way you make predictions.
EDIT: (thx to @baruuum for pointing this out)
Weights could possibly enter the picture when you compute functions of the predictions (e.g. posterior mean). Then it could make sense to weigh the predictions (e.g. compute weighted mean). The weighted predictions themselves are still IMHO meaningless: if the unweighted prediction is that probability of success is 0.8 and the weight is 2, it doesn't make sense to say that the weighted probability is 1.6.
Also keep in mind that the posterior unweighted mean answers the question "What responses would I expect, if I got a new dataset with the same predictors?" while the weighted mean answers "What responses would I expect, if I got a new dataset with the same predictors and the same weights?" Which question is more interesting depends on what your weights actually represent and your research question.