I believe I figured this out...
The main problem was that I didn't want to write a macro for this button and have that macro saved into all my spreadsheets. So I wrote the macro in my personal macro book and pointed a custom button to that macro. It turns out, the macro does not get saved to the "working" workbook, which is the desired effect.
Here is the code (recorded):
Sub ClearClipboard()
'
' ClearClipboard Macro
'
' Keyboard Shortcut: Ctrl+Shift+C
'
Application.CutCopyMode = False
End Sub
That did it, thanks!