Have you tried making the cups of dark chocolate, then painting the white chocolate stripes on the outside after? (you might need to paint several layers depending on how thick it is)
If the chocolate is mixing, then the second layer is remelting the first layer - this might be because of relative melting points, so the temperature needed to keep the second layer liquid is just too much higher than the melting point of the first, or it might be because of volume - adding a thick layer over a thinner is much more likely to melt because a thicker layer just has more heat.
So you can switch the order (first the cup, then the stripes) or the layer (paint the cup with the dark and the second layer is white, or the reverse of what you tried) to try and fix the problem mechanically.
Other things you might try include making the whole cup much colder between layers so that the second coat cools faster and doesn't have as much extra heat to melt the first, or just painting on many very thin layers (each layer would then have less heat to waste, cool faster, and give you more control over the design and thickness). This would be more work, though.