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To answer your specific question:
I am wondering if there is a way to send the context menu command to the application without having to have the context menu appear.
When a menu item is clicked, it issues a WM_COMMAND message to the parent window of the menu. You can use a tool like Spy++ to monitor such messages and see which command ID is being sent to which window, and then you can code your app to send the same message directly to that same window. No need to display the actual menu itself.
However, in this particular situation, that is overkill. If you have the HWND of the desired edit control, you can send it WM_GETTEXT and WM_SETTEXT messages instead, or use .NET's UIAutomation interfaces, to get/set the control's text directly. Either way, there is no need to select the text (you can use EM_SETSEL for that) or to involve the clipboard.