The menu function in the Content View will only produce a simple mean value, so you should use a formula for weighted means. You could achieve this by using the Table View and Aggregation, that will enable you to pick the weight field.

If you wish to visualise weighted means in the Bar/Line view, you could aggregate on the split field and choose weighted mean for the measure field.

Stacking the the averages usually doesn't visually work, because you end up with sum of group averages. It might be better if you add split by the grouping criteria, so you can compare the group means next to each other.

You can check the results by looking at the Table View, aggregating on the split field(s) and using weighted mean for the value field.