Export a layer in multiple colors

The way I would tackle that, is by using Layer comps:

Now the concept is a bit tricky to explain in text form, so I  am adding the link to the official help files:
https://helpx.adobe.com/uk/photoshop/using/layer-comps.html

to keep it short, layer comps are a way to "record states of your layer panel" and you can then take these different states and save each iteration. Recording in effect what is visible in your layer stack.

A good example of different layer comps would be

state 1: logo1 with red overlay (2 layers)
state 2: logo 1 with yellow overlay (2 layers)

state 3 :logo 3 with red overlay. (2 layers)

You got 3 different states.
To record those different states, you need to make visible only what you need in the layer stack, and click on the layer comp panel to create a new layer comp (a new state).

You can then export these different states as sepaparate files using the File>Export> layer comps to files

see an example for a commercial project of mine

each of those layer comp entry represent a set of visible layers (50 in total, 2 visible each time) : 1 is a smart object layer containing some text and another containing the label colours (these were labels placed on a mock up). For each layer comp, there was only 2 layers showing among the vast number of layers in the layer stack.

Hope it makes sense.

https://www.take-a-screenshot.org

I would expect that the best approach would be a custom Script. 

How familiar are you with JavaScript? 

But maybe you can also get some mileage out of data-driven graphics: 

Creating data-driven graphics in Photoshop - Adobe Supporthttps://helpx.adobe.com › photoshop › using › creating...

What is the Blend Mode of the coloring layers? 

The names of the coloring layers seem to be the default names, for full automation it might be better if the names make sense and can be used in naming the resulting files.