You do also need to have it act such as mouse. Since if you plug two Real Mouse in, only primary / first Mouse will work. If you Arduino its the cheat device Mouse, it will be plugged in first, you will not be able to click through menu or click play with your Real Mouse. You allowed one HID.
Personally I assert that if you're already running a model to classify pixels on the screen to try to aim, passing through mouse movements from another mouse in addition isn't really a big deal.
Most people lazy for this