I noticed that this is still on the 'Unanswered' page... So I decided to tackle it. I love figuring things out; we can do this!
I found that it's just simple math in the end – but unfortunately Illustrator is lacking the feature we need in this case.
Here are the two colors you used (on the left):
Color 1 is the base color and Color 2 is above it. As you've seen, applying a Blending Mode of Multiply to these two colors gives you this resulting third color:
This clearly does not give you the result you want, so as you've already deduced your second color is wrong.
To "reverse engineer" two colors – where it is known that Color 2 is the result of color multiplication – just take the two final colors and do the reverse.
Here's the math part:
(Using numerals for example) we know that 4 multiplied by 6 gives us 24
We also know that 24 divided by 6 gives us 4
(Now with colors) we know that Color 1 multiplied by Color 2 gives us Color 3
Therefore Color 3 divided by Color 2 will give us Color 1
Unfortunately for whatever reason (I don't know) Illustrator doesn't even have a Divide Blending Mode.
But Photoshop does...
So starting again in Photoshop now, using the two colors you want to end up with (as seen on the right in your example)... As per the example above we have Color 1 on top of Color 3:
Then change the Blending Mode of Color 1 to Divide, and the result gives us (the desired) Color 2:
A new color... This is the color you'll want to replace Color 2 (in the first example) with...
So now beginning again with our (new) Colors 1 and 2:
By applying the Multiply Blending Mode we now get the proper outcome:
I know I'm probably too late to help Alexander but this is useful information anyway... so here it is.