A possible way of doing this without too much effort would be to create a batch script which will start TeamViewer minimized.
Open up a blank notepad and enter the following (PATH_TO_TEAMVIEWER_EXECUTABLE - change this to the target which is set for your original TeamViewer shortcut in your startup folder) -
start /b /min "PATH_TO_TEAMVIEWER_EXECUTABLE"
Save this as TeamViewer.bat, or call it something else but it must have the ".bat" extension at the end. Change "Save as type" from "Text Documents (*.txt)" to "All Files (*.)". When you have completed those steps, click on "Save".
Now locate the batch file in which you just created, right-click on this file and click on "Copy". Open up your "Startup" folder, right-click...