So I had the same issue with another multi coloured logo I make.
Could not get it to work messing with the CAD design so I`ve spent a few hours just clicking buttons in the slicer.
Found the button that works.
View all objects setting tab to the right of the advanced tab. clicking on this bought up all the parts of the design and allowed me to choose the colour of each, without using the paint function.
Worked perfectly and all the parts are coloured to the depth I wanted.