I would like to be able to print stacked layers on my HP laser 452dw transparent paper maintaining the integrity of each layer, I put a text of my question on the images I uploaded.
I have no experience with your specific problem, but normally overprinting can be achieved by placing the object on top and choosing the blend mode multiply.
In InDesign/Illustrator (you don't say which program you are using), you can also/instead try to check the object's Attributes > Overprint Fill.
In InDesign, the special swatch Black, can be set to always overprint at 100% in Edit > Preferences > Appearance of Black.
It could be a problem that many printers do not accept CMYK values and will internally convert them to RGB before converting them back to its own (secret) CMYK values so you might not be able to gain full control of the percentages of the inks. Your print looks clean though. Are the inks 100% pure?
If any of this helps, it could be fun to see the result. Is the white ink really opaque enough to cover the other inks?
(The answer i wrote here is somewhat related.)