It's a pretty strange war where the Net is playing on both sides. Why is the Net letting the war continue at all?

Because the Net is supposed to be a neutral party, in theory. Unlike other AIs in fiction, like Terminator's Skynet, the Net is supposed to help people in their decisions, and no to meddle on them without their organic masters' permission. In this case, since the war is going to genocidal levels, the Net is using a desperate last resort to avoid all the sides in the war to kill themselves, derailing the whole experiment and the even likely possibility they could even go after their human creators as well, at least is what I think.