wormwood, thinking about what he has done so far and what he said in his manifesto, I don't think he cares about public opinion or trying to win over their support. I think he's looking at it a different way which is what we've already seen as you pointed out. Making the cops so scared that they, in turn, end up killing more innocent people than him. He's got them so trigger happy now that all he has to do is keep pushing the buttons at the right time and I wouldn't be surprised if the cop killing bodycount increases.
Right now the majority of people are against him but let the LAPD and other law enforcement agencies kill more people in their zeal to get him first and that will gradually change. If anything, the longer he can go on, the worse it's going to look for the cops. When that point happens, then the real investigations are going to start, not because of public opinion.
Here's another thing that people are arguing about is when he killed the cop's daughter and fiancee and people are shocked and condemning him. Again, if his goal isn't a PR campaign, by saying AND targeting family members, he accomplishes something else - not just a cop fearing his own life, but now fearing for the life of his family also. That weakens them. That lessens their advantage. Remember, he is waging war against the LAPD and he is going to take any advantage to win if winning is his primary goal.
Call it psychotic or demented or whatever, but those go hand in hand in waging war don't they?