Does the GPS antenna use a Fakra connector? How about using a relayed switch to disconnect the antenna. I saw this https://synergytradeshop.com/sw-v2x1-fak/ and am thinking plug the GPS antenna into 1 of the coax inputs and just leave the other one empty. Then take the output and send it to the 4400NEX. You could have another relay that defaults to closed when not powered for the VSS wire so it stays connected most of the time.
This would allow Android phones to scroll menus in Android Auto without the 6 click limit while stopped because the GPS will be hooked up. Also provides the better GPS signal through the head unit most of the time. For times you want to use video while in motion flick a dash mounted toggle switch. The switch would have accessory power so you don't draw with the car turned off. When the toggle is closed it would send power to both the coax relay switch linked above and the VSS relay. This would switch the VSS off and the GPS antenna off at the same time.
Another idea I had floating around was run the video output from the rear source video output on the head unit into one of the camera inputs with a patch cable. Select a video source to play to the rear. Then bring up the camera view on the head unit.