How to map CTRL + ALT + TAB to a macro activated from a customizable special key on the Microsoft Natural Ergonomic Keyboard 4000?

Using exactly the same keyboard model, I can easily re-map ⏵/⏸ (Play/Pause) button in AutoHotKey like this:

Media_Play_Pause::Send ^!{Tab}
+Media_Play_Pause::Media_Stop
^Media_Play_Pause::Launch_Media

...and your Ctrl+Alt+Tab idea works nicely. So does the rest of examples where I am demonstrating use of Shift and Ctrl modifiers.

But I admit I do not have Microsoft Mouse and Keyboard Center installed to avoid any interference (and to save some system resources).

For sake of completeness of this answer, I need to note that I am using the following modification to Send mode at the top of the AHK script:

SendMode Input

And make sure that you always run the AHK elevated (i. e. As Administrator).

I verified that the following script catches ⏵/⏸ (Play/Pause) button nicely in Windows 10, AHK (version 1.1.25.01) running elevated:

SendMode Input
+Media_Play_Pause::msgbox Shift+PlayPause
^Media_Play_Pause::msgbox Ctrl+PlayPause
Media_Play_Pause::msgbox PlayPause

After your further research, the problem was not in capturing ⏵/⏸ (Play/Pause) button but in executing of Alt+Tab which needed special technique for Windows 8.1. Capture itself went just fine.