How can I use the keyboard to switch the mouse between left-handed and right-handed?

Another AutoHotkey approach:

^!s::
run main.cpl
RegRead, swapped, HKEY_CURRENT_USER, Software\Synaptics\SynTPCpl\TouchPadPS2_3, SwapMouseButtons
WinWait,Mouse Properties
if (swapped = 1)
{
    RegWrite, REG_DWORD, HKEY_CURRENT_USER, Software\Synaptics\SynTPCpl\TouchPadPS2_3, SwapMouseButtons, % !swapped
    Send {TAB}{R}{A}{Enter}

}
else
{
    RegWrite, REG_DWORD, HKEY_CURRENT_USER, Software\Synaptics\SynTPCpl\TouchPadPS2_3, SwapMouseButtons, % !swapped
    Send {TAB}{L}{A}{Enter}
}
return

Ctrl + Alt + s will toggle the button swap.

After installing AutoHotkey, run it; you'll get a dialog to make a script:

Hit Yes and paste the above code into the file, save it. Now run AutoHotkey again so it sits in your tray. It will now listen for the key combinations above until you right-click its tray icon and exit it.

Sounds like a job for Autohotkey.

Something like:

^!c::
run main.cpl
WinWait ahk_class #32770
Send {Tab3}{Left}{Tab}{Space}{Enter}
return

To have Ctrl + Alt + C make the change. You'll need to adapt it a little for your set up.

Edit- This is closer, but still not quite perfect. My Dell touchpad utility might be getting in the way. Thanks for tips people

I have both a right handed and a Logitech left-handed mouse installed. The left-handed mouse is automatically configured backwards. When you use it, the buttons are reversed, so that it operates correctly. No mucking with control panel is required!