While 4 of the 8 wires are all that is needed to make the camera work yes.. The fact is that the camera is wired in a way that would require you to wire it as setup or the camera will fail if not connected to the right system!
Can I ask how you know for sure that the Green white wire is no longer working?
Anyway, The best thing to do is find someone that has the same camera if they would be willing to tone out the wiring from the camera ethernet to the inside connector. The issue is at times the camera could be short sourced and the wiring that is used isn't the same color to say another one that was built before or after that LOT.. However the PCB connector to the Ethernet will be wired the same no matter what the color of wire is used..
Best way to tone out the wires for what pin is what is to use a wire that has only 1 RJ45 connected to the wire and the other end removed.. Normally 1, 2, 3 and 6 are needed for the camera to work. with you saying that the Green/White wire AKA pin 3 sure enough the camera won't work..
Here is what some people don't understand about POE, With the Amcrest using Mode A using 1, 2, 3 and 6 pins. the data and power share the same wires Positive voltage V+ runs on 1 and 2 the Negative voltage V- runs on 3 and 6. Other systems that work on Mode B need all 8 wires where 4 5 and 78 would be power and first 4 wires would be data..
What I would do, Is confirm the wire colors for the connector and replace it with a tool less key stone jack and just make sure that the cable is not exposed to the weather. I am talking about one like these. I only picked this one because they offer 6 for double what others are selling 2 for. Why get 1/3 for 1/2 the cost.
6 Pcs RJ45 Cat5E Network LAN Ethernet Keystone Jack Tool Less Snap In 8P8C Ivory | eBay