Possible to check if cell has background color?

This is a continuation of attempting to override background colors set by conditional formatting and potentially an XY problem. Basically, it doesn’t seem like there’s a built-in function to check whether a cell has a background color set (the first condition to check). If so, prevent it from being changed by conditional formatting of alternating row colors (the second condition); otherwise, apply row alternating colors on the no-fill background color.

In the example on that thread, the condition being checked is whether the cell has the value of a number. However, the data on my cell is arbitrary.

How would one create the first condition to check whether background color is set in order to short circuit the second condition if the background color is indeed set? As a last resort, would it be possible to give cells with a background color a user-defined attribute that can be checked for in the first condition?

Please take a look to this thread Calc sum cells values having some background color