As you noticed the option Multicolored works only if there is one expression. There is a workaround:
Click on the + sign in front of the expression, select Background Color and add an expression to calculate the color for the bar. If you have a small number of dimension values (up to 20) you could use color(fieldindex('MyDim1',MyDim1)). Replace MyDim1 with the name of the dimension you use in your chart.