This is rough prototype.
I haven't tested here, so may need work-arounds to the work-around.
The goal is can click anywhere on the thin strip to bring up the standard, expanded, Global Floating palette, as shown above.
Not necessary to click any specific icon, because they all do the same thing.
- Set up several identical macros to open the KM Global Floating Palette.
"Several" can be 3-4-5-6: up to you.
The number of macros determines height of the skinny palette.
.
- Use KM preferences to set color, size, etc. of the actual skinny palette.
Key is minimize "size" = 8.
[√] Icons only to keep it thin.
.
I'll keep my "thinking cap" handy until results are what you want.